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question_answer1)
If \[{{S}_{n}}\]denotes the sum of first n terms of an A.P. and \[\frac{{{S}_{3n}}-{{S}_{n-1}}}{{{S}_{2n}}-{{S}_{2n-1}}}=31\], then the value of n is
A)
21 done
clear
B)
15 done
clear
C)
16 done
clear
D)
19 done
clear
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question_answer2)
If x, y, z are real and \[4{{x}^{2}}+9{{y}^{2}}+16{{z}^{2}}-6xy-12yz-8zx=0\], then x, y, z are in
A)
A.P done
clear
B)
G.P. done
clear
C)
H.P. done
clear
D)
none of these done
clear
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question_answer3)
Consider the ten numbers \[ar,a{{r}^{2}},a{{r}^{3}},...a{{r}^{10}}\].If their sum is 18 and the sum the reciprocals is 6, then the product of these ten numbers is
A)
81 done
clear
B)
243 done
clear
C)
343 done
clear
D)
324 done
clear
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question_answer4)
If the sum of n terms of an A.P. is cn (n - 1), where \[c\ne 0\]then the sum of the squares of these term is
A)
\[{{c}^{n}}n{{(n+1)}^{2}}\] done
clear
B)
\[\frac{2}{3}{{c}^{2}}n(n-1)(2n-1)\] done
clear
C)
\[\frac{2{{c}^{2}}}{3}n(n+1)(2n+1)\] done
clear
D)
none of these done
clear
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question_answer5)
The largest term common to the sequences 1, 11, 21, 31 ,..to 100 terms and 31, 36, 41, 46,..to 100 terms is
A)
381 done
clear
B)
471 done
clear
C)
281 done
clear
D)
none of these done
clear
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question_answer6)
If x, 2y, 3z are in A.P. where the distinct numbers x, y, z are in G.P., then the common ratio of the G.P. is
A)
3 done
clear
B)
\[\frac{1}{3}\] done
clear
C)
2 done
clear
D)
\[\frac{1}{2}\] done
clear
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question_answer7)
If \[{{S}_{n}}\]denotes the sum of first n terms of an A.P. whose first term is a and \[{{S}_{nx}}/{{S}_{x}}\]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
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question_answer8)
If a, b, and c are in A.P., p, q, and r are in H.P., and ap, bq, and cr are in G.P., then \[\frac{p}{r}+\frac{r}{p}\]is equal to
A)
\[\frac{a}{c}-\frac{c}{a}\] done
clear
B)
\[\frac{a}{c}+\frac{c}{a}\] done
clear
C)
\[\frac{b}{q}+\frac{q}{b}\] done
clear
D)
\[\frac{b}{q}-\frac{q}{b}\] done
clear
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question_answer9)
If \[{{a}_{1}},{{a}_{2}},{{a}_{3}}....{{a}_{n}}\]are in H.P. and \[f(k)=\left( \sum\limits_{r=1}^{n}{{{a}_{r}}} \right)-{{a}_{k}}\]then \[\frac{{{a}_{1}}}{f(1)},\frac{{{a}_{2}}}{f(2)},\frac{{{a}_{3}}}{f(3)},...\frac{{{a}_{n}}}{f(n)}\]are in
A)
A.P done
clear
B)
G.P done
clear
C)
H.P done
clear
D)
none of these done
clear
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question_answer10)
a, b, c, d,\[\in {{R}^{+}}\] such that a, b, and c are in A.P. and b, c and , d are in H.P., then
A)
ab = cd done
clear
B)
ac = bd done
clear
C)
bc = ad done
clear
D)
None of these done
clear
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question_answer11)
if a, b, c are in A.P., then\[\frac{a}{bc},\frac{1}{c},\frac{2}{b}\] will be in
A)
A.P. done
clear
B)
G.P. done
clear
C)
H.P. done
clear
D)
none of these done
clear
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question_answer12)
The 15th terms of the series \[2\frac{1}{2}+1\frac{7}{13}+1\frac{1}{9}+\frac{20}{23}+...\]is
A)
\[\frac{10}{39}\] done
clear
B)
\[\frac{10}{21}\] done
clear
C)
\[\frac{10}{23}\] done
clear
D)
none of these done
clear
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question_answer13)
If \[{{x}_{1}},{{x}_{2}},.....{{x}_{20}}\]are in H.P. and \[{{x}_{1}},2,{{x}_{20}}\]are in G.P., then \[\sum\limits_{r=1}^{19}{{{x}_{r}}{{x}_{r+1}}}\]=
A)
76 done
clear
B)
80 done
clear
C)
84 done
clear
D)
none of these done
clear
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question_answer14)
Sum of the series \[\frac{2}{3}+\frac{8}{9}+\frac{26}{27}+\frac{80}{81}+...\]to 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
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question_answer15)
The sum of the series \[\frac{x}{1-{{x}^{2}}}+\frac{{{x}^{2}}}{1-{{x}^{4}}}+\frac{{{x}^{4}}}{1-{{x}^{8}}}+...\]to infinite terms, if \[\left| x \right|<1\], is
A)
\[\frac{x}{1-x}\] done
clear
B)
\[\frac{1}{1-x}\] done
clear
C)
\[\frac{1+x}{1-x}\] done
clear
D)
1 done
clear
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question_answer16)
Let \[S=\frac{4}{19}+\frac{44}{{{19}^{2}}}+\frac{444}{{{19}^{3}}}+...\]up to \[\infty \]. Then S is equal to
A)
40/9 done
clear
B)
38/81 done
clear
C)
36/171 done
clear
D)
none of these done
clear
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question_answer17)
The sum of 20 terms of the series whose rth term is given by \[T(n)={{(-1)}^{n}}\frac{{{n}^{2}}+n+1}{n!}\]is
A)
\[\frac{20}{19!}-2\] done
clear
B)
\[\frac{21}{20!}-1\] done
clear
C)
\[\frac{21}{20!}\] done
clear
D)
none of these done
clear
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question_answer18)
If a, b, c and d are in H.P. then
A)
\[{{a}^{2}}+{{c}^{2}}>{{b}^{2}}+{{d}^{2}}\] done
clear
B)
\[{{a}^{2}}+{{d}^{2}}>{{b}^{2}}+{{c}^{2}}\] done
clear
C)
\[ac+bd>{{b}^{2}}+{{c}^{2}}\] done
clear
D)
\[ac+bd>{{b}^{2}}+{{d}^{2}}\] done
clear
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question_answer19)
If \[{{a}_{1}},{{a}_{2}}....{{a}_{n}}\]are in H.P., then the expression \[{{a}_{1}}{{a}_{2}}+{{a}_{2}}{{a}_{3}}\]+...+\[{{a}_{n}}{{-}_{1}}{{a}_{n}}\]is equal to
A)
\[n({{a}_{1}}-{{a}_{n}})\] done
clear
B)
\[(n-1)({{a}_{1}}-{{a}_{n}})\] done
clear
C)
\[n{{a}_{1}}{{a}_{n}}\] done
clear
D)
\[(n-1){{a}_{1}}{{a}_{n}}\] done
clear
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question_answer20)
The sum of the series \[1+\frac{1}{4.2!}+\frac{1}{16.4!}+\frac{1}{64.6!}+...\]is
A)
\[\frac{e-1}{\sqrt{e}}\] done
clear
B)
\[\frac{e+1}{\sqrt{e}}\] done
clear
C)
\[\frac{e-1}{2\sqrt{e}}\] done
clear
D)
\[\frac{e+1}{2\sqrt{e}}\] done
clear
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question_answer21)
\[{{1}^{3}}-{{2}^{3}}+{{3}^{3}}-{{4}^{3}}+...+{{9}^{3}}=\] ________.
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question_answer22)
The value of \[{{2}^{1/4}}\times {{4}^{1/8}}\times {{8}^{1/6}}.....\infty \,\]is _______.
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question_answer23)
The sum to 50 terms of the series\[1+2\left( 1+\frac{1}{50} \right)+3{{\left( 1+\frac{1}{50} \right)}^{2}}+...\] is given by __________.
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question_answer24)
The sum of 0.2+0.004+0.00006+0.0000008+...to \[\infty \]
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question_answer25)
If A.M., G.M., and H.M., of the first and last terms of the series 100, 101, 102,...n-1, n are the terms of the series itself, then the value of n is (100<n\[\le \]500) ________.
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