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question_answer1) Let \[{{a}_{1}},{{a}_{2}},.........,{{a}_{30}}\] be an A.P., \[S=\sum\limits_{i=1}^{30}{{{a}_{i}}}\] and \[T=\sum\limits_{i=1}^{15}{{{a}_{(2i-1)}}.}\] If \[{{a}_{5}}=27\] and \[S-2T=75,\] then \[{{a}_{10}}\] is equal to
question_answer2) The sum of the following series \[1+6+\frac{9({{1}^{2}}+{{2}^{2}}+{{3}^{2}})}{7}+\frac{12({{1}^{2}}+{{2}^{2}}+{{3}^{2}}+{{4}^{2}})}{9}\] \[+\frac{15({{1}^{2}}+{{2}^{2}}+....+{{5}^{2}})}{11}+....\] up to 15 terms, is:
question_answer3) Let \[{{S}_{n}}=1+q+{{q}^{2}}+......+{{q}^{n}}\] and \[{{T}_{n}}=1+\left( \frac{q+1}{2} \right)+{{\left( \frac{q+1}{2} \right)}^{2}}+.....+{{\left( \frac{q+1}{2} \right)}^{n}}\] where q is a real number and \[q\ne 1.\] If \[^{101}{{C}_{1}}{{+}^{101}}{{C}_{2}}.{{S}_{1}}+.....{{+}^{101}}{{C}_{101}}.{{S}_{100}}={{\alpha }^{100}}{{T}_{100}},\]then \[\alpha \] is equal to
question_answer4) Let \[{{a}_{1}},{{a}_{2}},{{a}_{3}}.......\] be terms on A.P. If \[\frac{{{a}_{1}}+{{a}_{2}}+.........{{a}_{p}}}{{{a}_{1}}+{{a}_{2}}+.........+{{a}_{q}}}=\frac{{{p}^{2}}}{{{q}^{2}}},\] \[p\ne q,\] if \[\frac{{{a}_{6}}}{{{a}_{21}}}\] equals \[\frac{11}{k}\] then k is
question_answer5) Fifth term of a GP is 2, then the product of its 9 terms is
question_answer6) The value of \[\sum\limits_{k=1}^{10}{\left( \sin \frac{2k\pi }{11}+i\cos \frac{2k\pi }{11} \right)}\] is of the form \[{{i}^{k}},\] where k is smallest natural number, then k is
question_answer7) The value of \[{{2}^{1/4}}{{.4}^{1/8}}{{.8}^{1/16}}.....\infty \] is
question_answer8) The sum \[\sum\limits_{k=1}^{20}{k\frac{1}{{{2}^{k}}}}\] is equal to \[2-\frac{k}{{{2}^{19}}},\] then k is
question_answer9) If the sum and product of the first three terms in an A.P. are 33 and 1155, respectively, then absolute value of its 11th term is
question_answer10) If \[{{a}_{1}},{{a}_{2}},{{a}_{3}},.......{{a}_{n}}\] are in A.P. and \[{{a}_{1}}+{{a}_{4}}+{{a}_{7}}+.....+{{a}_{16}}=114,\] then \[{{a}_{1}}+{{a}_{6}}+{{a}_{11}}+{{a}_{16}}\] is equal to
question_answer11) In the four numbers first three are in G.P. and last three are in A.P. whose common difference is 6. If the first and last numbers are same, then first number will be
question_answer12) If arithmetic mean of a and b is \[\frac{({{a}^{n+1}}+{{b}^{n+1}})}{{{a}^{n}}+{{b}^{n}}},\]then the value of n is equal to
question_answer13) The number of common terms to the two sequences 17, 21, 25,...... 417 and 16, 21, 26,........ 466 is
question_answer14) If product of three terms of a GP is 216, and sum of their products taken in pairs is 156, then greatest term is
question_answer15) If \[y={{3}^{x-1}}+{{3}^{-x-1}}\] (x real), then the least value of y is
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