JEE Main & Advanced Mathematics Sequence & Series Question Bank

done Self Evaluation Test - Sequences and Series Total Questions - 100

• question_answer1)
  Let \[{{A}_{n}}\] be the sum of the first n terms of the geometric series \[704+\frac{704}{2}+\frac{704}{4}+\frac{704}{8}+......\]and \[{{B}_{n}}\] be the sum of the first n terms of the geometric series \[1984+\frac{1984}{2}+\frac{1984}{4}+\frac{1984}{8}+......\] If \[{{A}_{n}}={{B}_{n}},\]then the value of n is (where \[n\in N\]).
  A)
  4 done clear
  B)
  5 done clear
  C)
  6 done clear
  D)
  7 done clear
  View Solution play_arrow

• question_answer2)
  If \[\left| \,r\, \right|>1\] and \[x=a+\frac{a}{r}+\frac{a}{{{r}^{2}}}+....to\,\,\infty \],\[y=b-\frac{b}{r}+\frac{b}{{{r}^{2}}}-....\,to\,\,\infty \]and \[z=c+\frac{c}{{{r}^{2}}}+\frac{c}{{{r}^{4}}}+....to\,\,\infty \]then \[\frac{xy}{z}=\]
  A)
  \[\frac{ab}{c}\] done clear
  B)
  \[\frac{ac}{b}\] done clear
  C)
  \[\frac{bc}{a}\] done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer3)
  ABC is a right angled triangle in which \[\angle B=90{}^\circ \]and \[BC=a\]. If n points \[{{L}_{1}},\,{{L}_{2}},....{{L}_{n}}\] on AB are such that AB is divided in n + 1 equal parts and \[{{L}_{1}}{{M}_{1}},\,\,{{L}_{2}}{{M}_{2}},....,\,\,{{L}_{n}}{{M}_{n}}\] are line segments parallel to BC and \[{{M}_{1}},\,{{M}_{2}},...{{M}_{n}}\] are on AC, then the sum of the lengths of \[{{L}_{1}}{{M}_{1}},{{L}_{2}}{{M}_{2}},....{{L}_{n}}{{M}_{n}}\] is
  A)
  \[\frac{a(n+1)}{2}\] done clear
  B)
  \[\frac{a(n-1)}{2}\] done clear
  C)
  \[\frac{an}{2}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer4)
  The sum of \[i-2-3i+4...\] up to 100 terms, where \[i=\sqrt{-1}\] is
  A)
  \[50(1-i)\] done clear
  B)
  \[25i\] done clear
  C)
  \[.25\,(1+i)\] done clear
  D)
  \[100(1-i)\] done clear
  View Solution play_arrow

• question_answer5)
  Consider the sequence \[8A+2B,\text{ }6A+B,\text{ }4A,\,\,2A-B,........\]  Which term of this sequence will have a coefficient of A which is twice the coefficient of B?
  A)
  10th done clear
  B)
  14th done clear
  C)
  16th done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer6)
  If a, b, c are positive numbers, then least value of\[(a+b+c)\left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right)\] is
  A)
  1 done clear
  B)
  6    done clear
  C)
  9 done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer7)
  If \[a=1+(\sqrt{3}-1)+{{(\sqrt{3}-1)}^{2}}+{{(\sqrt{3}-1)}^{3}}+....\] and\[ab=1\], then a and bare the roots of the equation
  A)
  \[{{x}^{2}}+4x-1=0\] done clear
  B)
  done clear
  C)
  \[{{x}^{2}}+4x+1=0\] done clear
  D)
  \[{{x}^{2}}-4x+1=0\] done clear
  View Solution play_arrow

• question_answer8)
  Let \[{{a}_{n}}\] be the nth term of an A.P. If \[\sum\limits_{r=1}^{100}{{{a}_{2r}}=\alpha }\] and \[\sum\limits_{r=1}^{100}{{{a}_{2r-1}}=\beta }\], then the common difference of the A.P. is
  A)
  \[\alpha -\beta \] done clear
  B)
  \[\beta -\alpha \] done clear
  C)
  \[\frac{\alpha -\beta }{2}\] done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer9)
  If \[x{{.}^{\ell n\left( \frac{y}{z} \right)}}.{{y}^{_{^{\ell n{{(XZ)}^{2}}}}}}.\,{{z}^{\ell n\left( \frac{x}{y} \right)}}={{y}^{4\,\ell n\,y}}\] for any x > 1, y >1 and z > 1, then which one of the following is correct?
  A)
  \[\ell n\,y\] is the GM of \[\ell n\,x,\,\ell n\,x,\,\ell n\,x\] and \[\ell n\,z\] done clear
  B)
  \[\ell n\,y\] is the AM of \[\ell n\,x,\,\ell n\,x,\,\ell n\,x\]and \[\ell n\,z\] done clear
  C)
  \[\ell n\,y\] is the HM of \[\ell n\,x,\,\ell n\,x,\,\ell n\,x\] and \[\ell n\,z\] done clear
  D)
  \[\ell n\,y\] is the AM of \[\ell n,\,\,In\,\,x,\,\,\ell n\,z\] and \[\ell n\,z\] done clear
  View Solution play_arrow

• question_answer10)
  What is the greatest value of the positive integer n satisfying the condition \[1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+....+\frac{1}{{{2}^{n-1}}}<2-\frac{1}{1000}\]?
  A)
  8 done clear
  B)
  9 done clear
  C)
  10 done clear
  D)
  11 done clear
  View Solution play_arrow

• question_answer11)
  If \[(1-p)(1+3x+9{{x}^{2}}+27{{x}^{3}}+81{{x}^{4}}+243{{x}^{5}})\]\[=1-{{p}^{6}},\] \[p\ne 1\] then the value of \[\frac{p}{x}\] is
  A)
  \[\frac{1}{3}\] done clear
  B)
  3 done clear
  C)
  \[\frac{1}{2}\] done clear
  D)
  2 done clear
  View Solution play_arrow

• question_answer12)
  Given that \[\alpha ,\,\gamma \]are root of the equation\[A{{x}^{2}}-4x+1=0\]. and \[\beta ,\,\delta \] the roots of the equation \[B{{x}^{2}}-6x+1=0,\] the values of A and B such that \[\alpha ,\beta ,\gamma \] and \[\delta \] are in H. P. are
  A)
  A = 3, B = 8 done clear
  B)
  A = -3, B = 8 done clear
  C)
  A = 3, B = - 8 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  If the pth  term of an A.P. \[be\,\,\frac{1}{q}\] and qth term be \[\frac{1}{p},\] then the sum of its pqth terms will be
  A)
  \[\frac{pq-1}{2}\] done clear
  B)
  \[\frac{1-pq}{2}\] done clear
  C)
  \[\frac{pq+1}{2}\] done clear
  D)
  \[-\frac{pq+1}{2}\] done clear
  View Solution play_arrow

• question_answer14)
  Sum of n terms of series \[12+16+24+40+...\]will be
  A)
  \[2({{2}^{n}}-1)+8n\] done clear
  B)
  \[2({{2}^{n}}-1)+6n\] done clear
  C)
  \[3({{2}^{n}}-1)+8n\] done clear
  D)
  \[4({{2}^{n}}-1)+8n\] done clear
  View Solution play_arrow

• question_answer15)
  If mth terms of the series 63+65+67+69+........... and 3+10+17+24+..........be equal, then m=
  A)
  11 done clear
  B)
  12    done clear
  C)
  13 done clear
  D)
  15 done clear
  View Solution play_arrow

• question_answer16)
  \[{{\left( x+\frac{1}{x} \right)}^{2}}+{{\left( {{x}^{2}}+\frac{1}{{{x}^{2}}} \right)}^{2}}+{{\left( {{x}^{{}}}+\frac{1}{{{x}^{3}}} \right)}^{2}}....\] upto n terms is
  A)
  \[\frac{{{x}^{2n}}-1}{{{x}^{2}}-1}\times \frac{{{x}^{2n+2}}+1}{{{x}^{2n}}}+2n\] done clear
  B)
  \[\frac{{{x}^{2n}}+1}{{{x}^{2}}+1}\times \frac{{{x}^{2n+2}}-1}{{{x}^{2n}}}-2n\] done clear
  C)
  \[\frac{{{x}^{2n}}-1}{{{x}^{2}}-1}\times \frac{{{x}^{2n}}-1}{{{x}^{2n}}}-2n\] done clear
  D)
  None of these done clear
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• question_answer17)
  Let \[{{S}_{n}}(1\le n\le 9)\] denotes the sum of n terms of series 1+22+333+.................+9999999999, then for \[2\le n\le 9\]
  A)
  \[{{S}_{n}}-{{S}_{n-1}}=\frac{1}{9}({{10}^{n}}-{{n}^{2}}+n)\] done clear
  B)
  \[{{S}_{n}}=\frac{1}{9}({{10}^{n}}-{{n}^{2}}+2n-2)\] done clear
  C)
  \[9\,({{S}_{n}}-{{S}_{n-1}})=n\,({{10}^{n}}-1)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If \[{{a}^{2}},{{b}^{2}},{{c}^{2}}\] are in A.P. consider two statements

  (i) \[\frac{1}{b+c},\frac{1}{c+a},\frac{1}{a+b}\] are in A.P.

  (ii) \[\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}\] are in A.P.

  A)
  (i) and (ii) both correct done clear
  B)
  (i) and (ii) both incorrect done clear
  C)
  (i) correct (ii) incorrect done clear
  D)
  (i) incorrect (ii) correct done clear
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• question_answer19)
  The value of the infinite product \[{{6}^{\frac{1}{2}}}\times {{6}^{\frac{1}{2}}}\times {{6}^{\frac{3}{8}}}\times {{6}^{\frac{1}{4}}}....\] is
  A)
  6 done clear
  B)
  36 done clear
  C)
  216 done clear
  D)
  \[\infty \] done clear
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• question_answer20)
  a, b, c are in G.P. with 1<a<b<n, and n>1 is an integer, \[lo{{g}_{a}}\text{ }n,\text{ }lo{{g}_{b}}\text{ }n,\text{ }lo{{g}_{c}}\,n\] form a sequence. This sequence is which one of the following?
  A)
  Harmonic progression done clear
  B)
  Arithmetic progression done clear
  C)
  Geometric progression done clear
  D)
  None of these done clear
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• question_answer21)
  If S, P and R are the sum, product and sum of the reciprocals of n terms of an increasing GP respectively and \[{{S}^{n}}={{R}^{n}}.{{P}^{k}},\] then k is equal to
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer22)
  If a, b and c are in H. P then the value of\[\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\left( \frac{1}{c}+\frac{1}{a}-\frac{1}{b} \right)\] is:
  A)
  \[\frac{2}{bc}+\frac{1}{{{b}^{2}}}\] done clear
  B)
  \[\frac{3}{{{c}^{2}}}+\frac{2}{ca}\] done clear
  C)
  \[\frac{3}{{{b}^{2}}}-\frac{2}{ab}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer23)
  A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of terms occupying odd places, then the common ratio is
  A)
  5 done clear
  B)
  1 done clear
  C)
  4 done clear
  D)
  3 done clear
  View Solution play_arrow

• question_answer24)
  The sum to n terms of the series 2+5+14+41+........is
  A)
  \[{{3}^{n-1}}+8n-3\] done clear
  B)
  \[{{8.3}^{n}}+4n-8\] done clear
  C)
  \[{{3}^{n+1}}+\frac{8}{3}n+1\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer25)
  If the coefficients of rth, (r + 1)th, and (r + 2)th terms in the binomial expansion of \[{{(1+y)}^{m}}\] are in A.P, then m and r satisfy the equation
  A)
  \[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}-2=0\] done clear
  B)
  \[{{m}^{2}}-m\left( 4r+1 \right)+4{{r}^{2}}+2=0\] done clear
  C)
  \[{{m}^{2}}-m(4r+1)+4{{r}^{2}}-2=0\] done clear
  D)
  \[{{m}^{2}}-m\left( 4r-1 \right)+4{{r}^{2}}+2=0\] done clear
  View Solution play_arrow

• question_answer26)
  If a, b, c are in A. P., then\[(a+2b-c)(2b+c-a)(c+a-b)\] equals
  A)
  \[\frac{1}{2}abc\] done clear
  B)
  abc done clear
  C)
  2 abc done clear
  D)
  4 abc done clear
  View Solution play_arrow

• question_answer27)
  If \[{{S}_{n}}=\]\[(1+{{3}^{-1}})(1+{{3}^{-2}})(1+{{3}^{-4}})(1+{{3}^{-8}})....(1+{{3}^{-{{2}^{n}}}}),\] then \[{{S}_{\infty }}\] is equal to
  A)
  1 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  \[\frac{3}{2}\] done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer28)
  If, 8, -4 and 13 be three (not necessarily consecutive term) of an A.P., how many such A.P. s are possible?
  A)
  1 done clear
  B)
  2 done clear
  C)
  Infinitely many done clear
  D)
  No such A.P. is possible done clear
  View Solution play_arrow

• question_answer29)
  What is the sum of the series\[1+\frac{1}{8}+\frac{1.3}{8.16}+\frac{1.3.5}{8.16.24}+....\infty \]?
  A)
  \[\frac{2}{\sqrt{3}}\] done clear
  B)
  \[2\sqrt{3}\] done clear
  C)
  \[\frac{\sqrt{3}}{2}\]      done clear
  D)
  \[\frac{1}{2\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer30)
  The value of \[0.0\overline{37}\] where \[0.0\overline{37}\] stands for the number .0373737........... is:
  A)
  37/1000 done clear
  B)
  37/990 done clear
  C)
  1/37 done clear
  D)
  1/27 done clear
  View Solution play_arrow

• question_answer31)
  If a, b, and c are in A.P ., and p and p' are, respectively, A.M. and G.M. between a and b while q, q' are, respectively, the A,M. and G M. between b and c, then
  A)
  \[{{p}^{2}}+{{q}^{2}}=p{{'}^{2}}+q{{'}^{2}}\] done clear
  B)
  \[pq=p'q'\] done clear
  C)
  \[{{p}^{2}}-{{q}^{2}}=p{{'}^{2}}-q{{'}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  ABCD is a square of lengths \[a,\,a\in N,a>1\]. Let \[{{L}_{1}},{{L}_{2}},{{L}_{3}},...\] be points BC such that \[B{{L}_{1}}={{L}_{1}}{{L}_{2}}={{L}_{2}}{{L}_{3}}=...=1\] and \[{{M}_{1}},{{M}_{2}},{{M}_{3}},...\] be points on CD such that \[C{{M}_{1}}={{M}_{1}}{{M}_{2}}={{M}_{2}}{{M}_{3}}=...=1\]. Then, \[\sum\limits_{n=1}^{a-1}{(AL_{n}^{2}+{{L}_{n}}M_{n}^{2})}\] is equal to
  A)
  \[\frac{1}{2}a{{(a-1)}^{2}}\] done clear
  B)
  \[\frac{1}{2}a(a-1)(4a-1)\] done clear
  C)
  \[\frac{1}{2}(a-1)(2a-1)(4a-1)\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer33)
  If \[{{S}_{n}}\] denotes the sum of first n terms of an A.P. whose first term is a and \[\frac{{{S}_{nx}}}{{{S}_{x}}}\] is independent of x, then \[{{S}_{p}}=\]
  A)
  \[{{P}^{3}}\] done clear
  B)
  \[{{P}^{2}}a\] done clear
  C)
  \[P{{a}^{2}}\] done clear
  D)
  \[{{a}^{3}}\] done clear
  View Solution play_arrow

• question_answer34)
  The value of x in \[(0,\pi )\] which satisfy the equation \[{{8}^{1+\left| \cos \,x \right|+xo{{s}^{2}}x+\left| {{\cos }^{3}}x \right|+.......to\,\,\infty \,}}={{4}^{3}}\] is
  A)
  \[\left\{ \frac{\pi }{2},\frac{3\pi }{4} \right\}\] done clear
  B)
  \[\left\{ \frac{\pi }{4},\frac{3\pi }{4} \right\}\] done clear
  C)
  \[\left\{ \frac{\pi }{3},\frac{2\pi }{3} \right\}\] done clear
  D)
  \[\left\{ \frac{\pi }{6},\frac{5\pi }{6} \right\}\] done clear
  View Solution play_arrow

• question_answer35)
  If \[{{a}_{1}},{{a}_{2}}.....,{{a}_{n}}\] are positive real numbers whose product is a fixed number c, then the minimum value of \[{{a}_{1}}+{{a}_{2}}+.....+{{a}_{n-1}}+2{{a}_{n}}\] is
  A)
  \[n{{(2c)}^{1/n}}\] done clear
  B)
  \[(n+1){{c}^{1/n}}\] done clear
  C)
  \[2n{{c}^{1/n}}\] done clear
  D)
  \[(n+1){{(2c)}^{1/n}}\] done clear
  View Solution play_arrow

• question_answer36)
  If \[\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}},\frac{1}{\sqrt{a}+\sqrt{b}}\] are in A.P. then \[{{9}^{ax+1}},{{9}^{bx+1}},{{9}^{cx+1}},x\ne 0\] are in:
  A)
  G.P done clear
  B)
  G.P. only if x<0 done clear
  C)
  G.P. only if x>0 done clear
  D)
  none of these done clear
  View Solution play_arrow

• question_answer37)
  If \[{{\log }_{10}}2,\,{{\log }_{10}}({{2}^{x}}-1)\] and \[{{\log }_{10}}({{2}^{x}}+3)\] are three consecutive terms of an A.P, then the value of x is
  A)
   1 done clear
  B)
  \[lo{{g}_{5}}2\] done clear
  C)
  \[lo{{g}_{2}}5\] done clear
  D)
  \[lo{{g}_{10}}5\] done clear
  View Solution play_arrow

• question_answer38)
  What is the 15th term of the series 3, 7, 13, 21, 31, 43, ....?
  A)
  205 done clear
  B)
  225 done clear
  C)
  238 done clear
  D)
  241 done clear
  View Solution play_arrow

• question_answer39)
  If a, b, c are in GP., then
  A)
  \[{{a}^{2}},\text{ }{{b}^{2}},\text{ }{{c}^{2}}\] are in GP. done clear
  B)
  \[{{a}^{2}}(b+c),{{c}^{2}}(a+b),{{b}^{2}}(a+c)\] are in G.P. done clear
  C)
  \[\frac{a}{b+c},\frac{b}{c+a},\frac{c}{a+b}\] are in G.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer40)
  If \[x=\sum\limits_{n=0}^{\infty }{{{a}^{n}}},y=\sum\limits_{n=0}^{\infty }{{{b}^{n}}},\,z=\sum\limits_{n=0}^{\infty }{{{c}^{n}}}\] where a, b, c are in A.P and \[\left| a \right|<1,\left| b \right|<1,\left| c \right|<1\] then \[x,y,z\] are in
  A)
  G.P. done clear
  B)
  A.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
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• question_answer41)
  The sum of the series \[\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+....n\] terms is:
  A)
  \[n-\frac{1}{2}({{3}^{n}}-1)\] done clear
  B)
  \[n+\frac{1}{2}({{3}^{n}}-1)\] done clear
  C)
  \[n-\frac{1}{2}(1-{{3}^{-n}})\] done clear
  D)
  \[n+\frac{1}{2}({{3}^{-n}}-1)\] done clear
  View Solution play_arrow

• question_answer42)
  The value of \[\sum\limits_{n=1}^{10}{\sum\limits_{m=1}^{10}{({{m}^{2}}+{{n}^{2}})}}\] equals
  A)
  4235 done clear
  B)
  5050 done clear
  C)
  7700 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer43)
  Concentric circles of radii 1, 2, 3,...100 cm are drawn. The interior of the smallest circle is coloured red and the angular regions are coloured alternately green and red, so that no two adjacent regions are of the same colour. The total area of the green regions on sq cm is equal to
  A)
  \[1000\,\pi \] done clear
  B)
  \[5050\,\pi \] done clear
  C)
  \[4950\,\pi \] done clear
  D)
  \[5151\text{ }\pi \] done clear
  View Solution play_arrow

• question_answer44)
  If \[{{a}_{1}},{{a}_{2}},{{a}_{3}}.....\] are in A.P. and \[a_{1}^{2}-a_{2}^{2}+a_{3}^{2}-a_{4}^{2}+......+a_{2k-1}^{2}-a_{2k}^{2}\] \[=M(a_{1}^{2}-a_{2k}^{2}).\] Then M =
  A)
  \[\frac{k-1}{k+1}\] done clear
  B)
  \[\frac{k}{2k-1}\] done clear
  C)
  \[\frac{k+1}{2k+1}\] done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer45)
  The maximum sum of the series \[20+19\frac{1}{3}\]\[+18\frac{2}{3}+18+....\] is
  A)
  300 done clear
  B)
  310 done clear
  C)
  \[311\frac{2}{3}\] done clear
  D)
  \[333\frac{1}{3}\] done clear
  View Solution play_arrow

• question_answer46)
  What is the sum of the series 0.5 + 0.55 + 0.555 +... to n terms?
  A)
  \[\frac{5}{9}\left[ n-\frac{2}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done clear
  B)
  \[\frac{1}{9}\left[ 5-\frac{2}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done clear
  C)
  \[\frac{1}{9}\left[ n-\frac{5}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done clear
  D)
  \[\frac{5}{9}\left[ n-\frac{1}{9}\left( 1-\frac{1}{{{10}^{n}}} \right) \right]\] done clear
  View Solution play_arrow

• question_answer47)
  If \[(1+3+5+...+p)+(1+3+5+...+q)\]\[=(1+3+5+...+r)\]where each set of parentheses contains the sum of consecutive odd integers as shown, what is the smallest possible value of (p + q + r) where\[p>6\]?
  A)
  12 done clear
  B)
  21 done clear
  C)
  45 done clear
  D)
  54 done clear
  View Solution play_arrow

• question_answer48)
  Three numbers are in G.P. such that their sum is 38 and their product is 1728. The greatest number among them is:
  A)
  18 done clear
  B)
  16 done clear
  C)
  14 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer49)
  The minimum value of \[\frac{{{x}^{4}}+{{y}^{4}}+{{z}^{2}}}{xyz}\] for positive real number x, y, z is
  A)
  \[\sqrt{2}\] done clear
  B)
  \[2\sqrt{2}\] done clear
  C)
  \[4\sqrt{2}\] done clear
  D)
  \[8\sqrt{2}\] done clear
  View Solution play_arrow

• question_answer50)
  \[\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{n(n+1)}\] equals
  A)
  \[\frac{1}{n(n+1)}\] done clear
  B)
  \[\frac{n}{n+1}\] done clear
  C)
  \[\frac{2n}{n+1}\] done clear
  D)
  \[\frac{2}{n(n+1)}\] done clear
  View Solution play_arrow

• question_answer51)
  For \[-\pi <x<\pi ,\] the values of x which satisfy the relation \[{{11}^{1+\left| \cos \,x \right|+co{{s}^{2}}x+\left| {{\cos }^{3}}x \right|+...upt{{o}^{\infty }}}}=121\] are given by
  A)
  \[\pm \frac{\pi }{3},\pm \frac{2\pi }{3}\] done clear
  B)
  \[\frac{\pi }{3},\frac{2\pi }{4}\] done clear
  C)
  \[\frac{\pi }{4},\frac{3\pi }{4}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer52)
  If the roots of the equation \[{{x}^{3}}-12{{x}^{2}}+39x-28=0\] are in A.P., then their common difference will be:
  A)
  \[\pm 1\] done clear
  B)
  \[\pm 2\]  done clear
  C)
  \[\pm 3\] done clear
  D)
  \[\pm 4\] done clear
  View Solution play_arrow

• question_answer53)
  There are four numbers of which the first three are in G.P. and the last three are in A.R, whose common difference is 6. If the first and the last numbers are equal then two other numbers are
  A)
  -2, 4 done clear
  B)
  -4, 2 done clear
  C)
  2, 6 done clear
  D)
  none done clear
  View Solution play_arrow

• question_answer54)
  It is given that \[\frac{1}{{{2}^{n}}\sin \,\alpha },1,\,{{2}^{n}}\sin \] a are in A.P. for some value of \[\alpha \]. Let say for n = 1, the \[\alpha \] satisfying the above A.P. is \[{{\alpha }_{1}},\] for n = 2, the value is \[{{\alpha }_{2}},\] and so on. If \[S=\sum\limits_{i=1}^{\infty }{\sin \,{{\alpha }_{i}},}\]then the value of S is
  A)
  1 done clear
  B)
  \[\frac{1}{2}\] done clear
  C)
  2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer55)
  The sum of an infinite GP is x and the common ratio r is such that \[\left| r \right|<1\]. If the first term of the GP is 2, then which one of the following is correct?
  A)
  \[-1<x<1\] done clear
  B)
  \[-\infty <x<1\] done clear
  C)
  \[1<x<\infty \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer56)
  Let a, b, c be in AP.

  Consider the following statements:

  1. \[\frac{1}{ab},\frac{1}{ca}\] and \[\frac{1}{bc}\]  are in A.P.

  2. \[\frac{1}{\sqrt{b}+\sqrt{c}},\frac{1}{\sqrt{c}+\sqrt{a}}\] and \[\frac{1}{\sqrt{a}+\sqrt{b}}\] are in A.P.

  Which of the statements given above is/are correct?

  A)
  1 only done clear
  B)
  2 only done clear
  C)
  Both 1 and 2      done clear
  D)
  Neither 1 and 2 done clear
  View Solution play_arrow

• question_answer57)
  lf \[x=1+a+{{a}^{2}}+....................\]to infinity and \[y=1+b+{{b}^{2}}+....................\] to infinity, where a, b are proper fractions, then \[1+ab+{{a}^{2}}{{b}^{2}}+.....\] to infinity is equal:
  A)
  \[\frac{xy}{x+y-1}\] done clear
  B)
  \[\frac{xy}{x-y-1}\] done clear
  C)
  \[\frac{xy}{x-y+1}\] done clear
  D)
  \[\frac{xy}{x+y+1}\] done clear
  View Solution play_arrow

• question_answer58)
  If \[a+b+c=3\] and \[a>0,b>0,c>0,\] then the greatest value of \[{{a}^{2}}{{b}^{3}}{{c}^{2}}\] is
  A)
  \[\frac{{{3}^{10}}{{.2}^{4}}}{{{7}^{7}}}\] done clear
  B)
  \[\frac{{{3}^{9}}{{.2}^{4}}}{{{7}^{7}}}\] done clear
  C)
  \[\frac{{{3}^{8}}{{.2}^{4}}}{{{7}^{7}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer59)
  If the sum to infinity of the series, \[1+4x+7{{x}^{2}}+10{{x}^{3}}+........,\] is 35/16, where \[\left| x \right|<1\], then x equals to
  A)
  19/7 done clear
  B)
  1/5 done clear
  C)
  ¼ done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer60)
  The sum of the series 3.6 + 4.7 + 5.8 + ......upto (n - 2) terms
  A)
  \[{{n}^{3}}+{{n}^{2}}+n+2\] done clear
  B)
  \[\frac{1}{6}(2{{n}^{3}}+12{{n}^{2}}+10\,n-84)\] done clear
  C)
  \[{{n}^{3}}+{{n}^{2}}+n\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer61)
  If the nth term of an arithmetic progression is\[3n+7\], then what is the sum of its first 50 terms?
  A)
  3925 done clear
  B)
  4100 done clear
  C)
  4175 done clear
  D)
  8200 done clear
  View Solution play_arrow

• question_answer62)
  A series is such that its every even term is 'a' times the term before it and every odd term is c times the term before it. The sum of 2n term of the series is (the first term is unity)
  A)
  \[\frac{(1-{{c}^{n}})(1-{{a}^{n}})}{1-ac}\] done clear
  B)
  \[\frac{(1+a)(1-{{c}^{n}}{{a}^{n}})}{1-ac}\] done clear
  C)
  \[\frac{(1+{{c}^{n}})(1+{{a}^{n}})}{1-ac}\] done clear
  D)
  \[\frac{(1+a)(1+{{c}^{n}}{{a}^{n}})}{1+ac}\] done clear
  View Solution play_arrow

• question_answer63)
  Let a = 111... 1(55 digits), \[b=1+10+{{10}^{2}}+...+{{10}^{4}},\] \[c=1+{{10}^{5}}+{{10}^{10}}+{{10}^{15}}+....+{{10}^{50}},\]then
  A)
  \[a=b+c\] done clear
  B)
  \[a=bc\] done clear
  C)
  \[b=ac\] done clear
  D)
  \[c=ab\] done clear
  View Solution play_arrow

• question_answer64)
  The value of x + y + z is 15 if a, x, y, z, b are in A.P. while the value of \[\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\]  is \[\frac{5}{3}\] if a, x, y, z, b are in H.P. Then the value of a and b are
  A)
  2 and 8 done clear
  B)
  1 and 9 done clear
  C)
  3 and 7 done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer65)
  If the positive integers a, b, c, d are in AP, then the numbers abc, abd, acd, bcd are in
  A)
  HP done clear
  B)
  AP done clear
  C)
  GP done clear
  D)
  None of the above done clear
  View Solution play_arrow

• question_answer66)
  Which one of the following options is correct?
  A)
  \[si{{n}^{2}}30{}^\circ ,\text{ }si{{n}^{2}}45{}^\circ ,\text{ }si{{n}^{2}}60{}^\circ \] are in GP done clear
  B)
  \[co{{s}^{2}}30{}^\circ ,\text{ }co{{s}^{2}}45{}^\circ ,\text{ }co{{s}^{2}}60{}^\circ \] are in GP done clear
  C)
  \[co{{t}^{2}}30{}^\circ ,\text{ }co{{t}^{2}}45{}^\circ ,\text{ }co{{t}^{2}}60{}^\circ \] are in GP done clear
  D)
  \[ta{{n}^{2}}30{}^\circ ,\text{ }ta{{n}^{2}}45{}^\circ ,\text{ }ta{{n}^{2}}60{}^\circ \] are in GP done clear
  View Solution play_arrow

• question_answer67)
  a, b, c are three distinct real numbers and they are in a GP. if a + b + c = xb, then
  A)
  \[x\le -1\,\,or\,\,x\ge 3\] done clear
  B)
  \[x<-1\,\,or\,\,x>3\] done clear
  C)
  \[x\le -1\,\,or\,\,x>3\] done clear
  D)
  \[x<-3\,\,or\,\,x>2\] done clear
  View Solution play_arrow

• question_answer68)
  If a, b, c are the sides of a triangle, then the minimum value of \[\frac{a}{b+c-a}+\frac{b}{c+a-b}+\]\[\frac{c}{a+b-c}\] is equal to
  A)
  3 done clear
  B)
  6     done clear
  C)
  9 done clear
  D)
  12 done clear
  View Solution play_arrow

• question_answer69)
  The 20th terms of the series 2 + 3 + 5 +9+16 +....... is
  A)
  950 done clear
  B)
  975   done clear
  C)
  990 done clear
  D)
  1010 done clear
  View Solution play_arrow

• question_answer70)
  The 100th term of the sequence 1, 2, 2, 3, 3, 3, 4, 4, 4, 4,... is
  A)
  12 done clear
  B)
  13    done clear
  C)
  14 done clear
  D)
  15 done clear
  View Solution play_arrow

• question_answer71)
  . An A.P. whose first term is unity and in which the sum of first half of any even number of terms to that of second half of the same number of terms is a constant ratio, then the common difference is:              
  A)
  2 done clear
  B)
  1 done clear
  C)
  3 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer72)
  The perimeter of a triangle whose sides are in A.P. is 21 cm and the product of lengths of the shortest side and the longest side exceeds the length of the other side by 6 cm. The longest side of the triangle is
  A)
  1 cm done clear
  B)
  7 cm done clear
  C)
  13 cm done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer73)
  The expression \[\frac{{{a}^{n+1}}+{{b}^{n+1}}}{{{a}^{n}}+{{b}^{n}}}\] is \[[a\ne b\ne 0]\] is (where a and b are unequal non-zero numbers)
  A)
  A.M. between a and b if \[n=-1\] done clear
  B)
  G.M. between a and b if \[n=-\frac{1}{2}\] done clear
  C)
  H.M. between a and b if n = 0 done clear
  D)
  All are correct done clear
  View Solution play_arrow

• question_answer74)
  The sum of \[\frac{\frac{1}{2}.\frac{2}{2}}{{{1}^{3}}}+\frac{\frac{2}{2}.\frac{3}{2}}{{{1}^{3}}+{{2}^{3}}}+\frac{\frac{3}{2}.\frac{4}{2}}{{{1}^{3}}+{{2}^{3}}+{{3}^{3}}}+.....\] upto n terms is equal to
  A)
              \[\frac{n-1}{n}\] done clear
  B)
  \[\frac{n}{n+1}\] done clear
  C)
              \[\frac{n+1}{n+2}\] done clear
  D)
  \[\frac{n+1}{n}\] done clear
  View Solution play_arrow

• question_answer75)
  What is the product of first 2n + 1 terms of a geometric progression?
  A)
  The (n + 1)th power of the nth term of the GP done clear
  B)
  The (2n + 1)th power of the nth term of the GP done clear
  C)
  The (2n + 1)th power of the (n + 1)th term of the GP done clear
  D)
  The nth power of the (n + 1)th terms of the GP done clear
  View Solution play_arrow

• question_answer76)
  If \[{{\log }_{10}}2,\,{{\log }_{10}}({{2}^{x}}-1),lo{{g}_{10}}({{2}^{x}}+3)\] are three consecutive terms of an AP, then which one of the following is correct?
  A)
  \[x=0\] done clear
  B)
  \[x=1\] done clear
  C)
  \[x={{\log }_{2}}5\] done clear
  D)
  \[x={{\log }_{5}}2\] done clear
  View Solution play_arrow

• question_answer77)
  Let S be the sum, P be the product and R be the sum of the reciprocals of 3 terms of a G.P. Then \[{{P}^{2}}{{R}^{3}}:{{S}^{3}}\] is equal to
  A)
  1 : 1 done clear
  B)
  Common ratio: 1 done clear
  C)
  \[{{\left( \text{first term} \right)}^{\text{2}}}\text{: }{{\left( \text{common ratio} \right)}^{\text{2}}}\] done clear
  D)
  \[{{\left( common\text{ }ratio \right)}^{n}}:1\] done clear
  View Solution play_arrow

• question_answer78)
  If \[{{t}_{n}}\] denotes the nth term of a G.P. whose common ratio is r, then the progression whose nth term is \[\frac{1}{t_{n}^{2}+t_{n+1}^{2}}\] is
  A)
  A.P. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer79)
  If \[{{\log }_{e}}5,\,{{\log }_{e}}({{5}^{x}}-1)\] and \[{{\log }_{e}}\left( {{5}^{x}}-\frac{11}{5} \right)\]are in A.P then the values of x are
  A)
  \[{{\log }_{5}}4\,\,and\,\,{{\log }_{5}}3\] done clear
  B)
  \[{{\log }_{3}}4\,\,and\,\,{{\log }_{4}}3\] done clear
  C)
  \[{{\log }_{3}}4\,\,and\,\,{{\log }_{3}}5\] done clear
  D)
  \[{{\log }_{5}}6\,\,and\,\,{{\log }_{5}}7\] done clear
  View Solution play_arrow

• question_answer80)
  2+4+7+11+16+........ to n terms =
  A)
  \[\frac{1}{6}({{n}^{2}}+3n+8)\] done clear
  B)
  \[\frac{n}{6}({{n}^{2}}+3n+8)\] done clear
  C)
  \[\frac{1}{6}({{n}^{2}}-3n+8)\] done clear
  D)
  \[\frac{n}{6}({{n}^{2}}-3n+8)\] done clear
  View Solution play_arrow

• question_answer81)
  If the angles A< B<C of a triangle are in A.P., then
  A)
  \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}-ab\] done clear
  B)
  \[{{b}^{2}}={{a}^{2}}+{{c}^{2}}-ac\] done clear
  C)
  \[{{c}^{2}}={{a}^{2}}+{{b}^{2}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer82)
  Let a, b, c, be in A.P. with a common difference d. Then \[{{e}^{1/c}},{{e}^{b/ac}},{{e}^{1/a}}\] are in:
  A)
  G.P. with common ratio \[{{e}^{d}}\] done clear
  B)
  G.P. with common ratio \[{{e}^{1/d}}\] done clear
  C)
  G.P. with common ratio \[{{e}^{d/({{b}^{2}}-{{d}^{2}})}}\] done clear
  D)
              A.P. done clear
  View Solution play_arrow

• question_answer83)
  The least value of n (a natural number), for which the sum S of the series \[1+\frac{1}{2}+\frac{1}{{{2}^{2}}}+\frac{1}{{{2}^{3}}}+.....\]differs from \[{{S}_{n}}\] by a quantity \[<{{10}^{-6}}\], is
  A)
  21 done clear
  B)
  20   done clear
  C)
  19 done clear
  D)
  None done clear
  View Solution play_arrow

• question_answer84)
  \[\sum\limits_{k=1}^{n}{k{{(1+1/n)}^{k-1}}=}\]
  A)
  \[n(n-1)\] done clear
  B)
  \[n(n+1)\] done clear
  C)
  \[{{n}^{2}}\] done clear
  D)
  \[{{(n+1)}^{2}}\] done clear
  View Solution play_arrow

• question_answer85)
  The equation \[\left( {{a}^{2}}+{{b}^{2}} \right){{x}^{2}}-2b\left( a+c \right)x+\] \[\left( {{b}^{2}}+{{c}^{2}} \right)=0\] has equal roots. Which one of the following is correct about a, b, and c?
  A)
  They are in AP done clear
  B)
  They are in GP done clear
  C)
  They are in HP done clear
  D)
  They are neither in AP, nor in GP, nor in HP done clear
  View Solution play_arrow

• question_answer86)
  What does the series \[1+{{3}^{-\frac{1}{2}}}+3+\frac{1}{3\sqrt{3}}+...\] represents?
  A)
  AP done clear
  B)
  GP done clear
  C)
  HP done clear
  D)
  None of the above series done clear
  View Solution play_arrow

• question_answer87)
  In a, GP. of 3n terms, \[{{S}_{1}}\] denotes the sum of first n terms, \[{{S}_{2}}\] the sum of the second block of n terms and \[{{S}_{3}}\] the sum of last n terms. Then \[{{S}_{1}},{{S}_{2}},{{S}_{3}},\] are in
  A)
  A.P. done clear
  B)
  G.P. done clear
  C)
  H.P. done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer88)
  If \[x>0,\frac{{{x}^{n}}}{1+x+{{x}^{2}}+...+{{x}^{2n}}}\] is
  A)
  \[\cdot \le \frac{1}{2n+1}\] done clear
  B)
  \[<\frac{1}{2n+1}\] done clear
  C)
  \[^{3}\frac{1}{2n+1}\] done clear
  D)
  \[>\frac{2}{2n+1}\] done clear
  View Solution play_arrow

• question_answer89)
  If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},.....{{a}_{n}}\] are in A.P., and\[\frac{1}{{{a}_{1}}{{a}_{n}}}+\frac{1}{{{a}_{2}}{{a}_{n-1}}}+\frac{1}{{{a}_{3}}{{a}_{n-2}}}+.....+\frac{1}{{{a}_{n}}{{a}_{1}}}\]\[=K\left( \frac{1}{{{a}_{1}}}+\frac{1}{{{a}_{2}}}+\frac{1}{{{a}_{3}}}+....+\frac{1}{{{a}_{n}}} \right)\]. Then K is
  A)
  \[\frac{2}{{{a}_{1}}+{{a}_{n}}}\] done clear
  B)
  \[\frac{n}{{{a}_{1}}+{{a}_{n}}}\] done clear
  C)
  \[\frac{1}{{{a}_{1}}+{{a}_{n}}}\] done clear
  D)
  \[\frac{n-1}{{{a}_{1}}+{{a}_{n}}}\] done clear
  View Solution play_arrow

• question_answer90)
  If the sum of the first ten terms of the series \[{{\left( 1\frac{3}{5} \right)}^{2}}+{{\left( 2\frac{2}{5} \right)}^{2}}+{{\left( 3\frac{1}{5} \right)}^{2}}+{{4}^{2}}+{{\left( 4\frac{4}{5} \right)}^{2}}+.....,\] is \[\frac{16}{5}m,\] then m is equal to:
  A)
  100 done clear
  B)
  99    done clear
  C)
  102 done clear
  D)
  101 done clear
  View Solution play_arrow

• question_answer91)
  If \[\frac{1}{a},\frac{1}{b},\frac{1}{c}\] are A. P., then \[\left( \frac{1}{a}+\frac{1}{b}-\frac{a}{c} \right)\] \[\left( \frac{1}{b}+\frac{1}{c}-\frac{1}{a} \right)\] is equal to
  A)
  \[\frac{4}{ac}-\frac{3}{{{b}^{2}}}\] done clear
  B)
  \[\frac{{{b}^{2}}-ac}{{{a}^{2}}{{b}^{2}}{{c}^{2}}}\] done clear
  C)
  \[\frac{4}{ac}-\frac{1}{{{b}^{2}}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer92)
  The roots of the equation \[{{(x-1)}^{2}}-4\left| x-1 \right|+3=0\]
  A)
  Form an A.P. done clear
  B)
  Form a G.P. done clear
  C)
  Form an H.P. done clear
  D)
  Do not form any progression done clear
  View Solution play_arrow

• question_answer93)
  x and y are positive number. Let g and a be G. M. and AM of these numbers. Also let G be G.M. of x + 1 and y + 1. If G and g are roots of equation\[{{x}^{2}}-5x+6=0\], then
  A)
  \[x=2,\,y=\frac{3}{4}\] done clear
  B)
  \[x=\frac{3}{4},y=12\] done clear
  C)
  \[x=\frac{5}{2},y=\frac{8}{5}\] done clear
  D)
  \[x=y=2\] done clear
  View Solution play_arrow

• question_answer94)
  The sum of the infinite series \[\frac{{{2}^{2}}}{2!}+\frac{{{2}^{4}}}{4!}+\frac{{{2}^{6}}}{6!}+....\] is equal to             
  A)
  \[\frac{{{e}^{2}}+1}{2e}\] done clear
  B)
  \[\frac{{{e}^{4}}+1}{2{{e}^{2}}}\] done clear
  C)
  \[\frac{{{({{e}^{2}}-1)}^{2}}}{2{{e}^{2}}}\] done clear
  D)
  \[\frac{{{({{e}^{2}}+1)}^{2}}}{2{{e}^{2}}}\] done clear
  View Solution play_arrow

• question_answer95)
  If \[\left| x \right|<\frac{1}{2},\] what is the value of \[1+n\left[ \frac{x}{1-x} \right]+\left[ \frac{n(n+1)}{2!} \right]{{\left[ \frac{x}{1-x} \right]}^{2}}+.......\infty \]?
  A)
  \[{{\left[ \frac{1-x}{1-2x} \right]}^{n}}\] done clear
  B)
  \[{{(1-x)}^{n}}\] done clear
  C)
  \[{{\left[ \frac{1-2x}{1-x} \right]}^{n}}\] done clear
  D)
  \[{{\left( \frac{1}{1-x} \right)}^{n}}\] done clear
  View Solution play_arrow

• question_answer96)
  The harmonic mean H of two numbers is 4 and the arithmetic mean A and geometric mean G satisfy the equation \[2A+{{G}^{2}}=27\]. The two numbers are
  A)
  6, 3 done clear
  B)
  9, 5 done clear
  C)
  12, 7 done clear
  D)
  3, 1 done clear
  View Solution play_arrow

• question_answer97)
  The sum of an infinite geometric series is 2 and the sum of the geometric series made from the cubes of this infinite series is 24. Then the series  is
  A)
  \[3+\frac{3}{2}-\frac{3}{4}+\frac{3}{8}-....\] done clear
  B)
  \[3+\frac{3}{2}-\frac{3}{4}+\frac{3}{8}+....\] done clear
  C)
  \[3-\frac{3}{2}+\frac{3}{4}-\frac{3}{8}+....\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer98)
  a, b, c are the first three terms of a geometric series. If the harmonic mean of a and b is 12 and that of b and c is 36, then the first five terms of the series are
  A)
  9, 18, 27, 36, 45 done clear
  B)
  8, 24, 72, 216, 648 done clear
  C)
  4, 22, 38, 46 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer99)
  The fourth term of an A.P. is three times of the first term and the seventh term exceeds the twice of the third term by one, then the common difference of the progression is
  A)
  2 done clear
  B)
  3 done clear
  C)
  \[\frac{3}{2}\] done clear
  D)
  -1 done clear
  View Solution play_arrow

• question_answer100)
  The sum to n terms of the series \[\frac{1}{2}+\frac{3}{4}+\frac{7}{8}+\frac{15}{16}+..........\] is
  A)
              \[n-1-{{2}^{-n}}\] done clear
  B)
  1 done clear
  C)
              \[n-1+{{2}^{-n}}\] done clear
  D)
  \[1+{{2}^{-n}}\] done clear
  View Solution play_arrow