Sequences and Series

\(x = \sum\limits_{n = 0}^\infty {{a^n},y = \sum\limits_{n = 0}^\infty {{b^n},z = \sum\limits_{n = 0}^\infty {{c^n}} } }\), where a, b, c are in A.P. and |a| < 1, |b| < 1, |c| < 1, abc \(\ne\) 0, then :

If a1, a2, a3 ...... and b1, b2, b3 ....... are A.P., and a1 = 2, a10 = 3, a1b1 = 1 = a10b10, then a4 b4 is equal to -

Let \(S = 2 + {6 \over 7} + {{12} \over {{7^2}}} + {{20} \over {{7^3}}} + {{30} \over {{7^4}}} + \,.....\). Then 4S is equal to

Suppose \(a_{1}, a_{2}, \ldots, a_{n}\), .. be an arithmetic progression of natural numbers. If the ratio of the sum of first five terms to the sum of first nine terms of the progression is \(5: 17\) and , \(110 < {a_{15}} < 120\), then the...

\(\frac{2^{3}-1^{3}}{1 \times 7}+\frac{4^{3}-3^{3}+2^{3}-1^{3}}{2 \times 11}+\frac{6^{3}-5^{3}+4^{3}-3^{3}+2^{3}-1^{3}}{3 \times 15}+\cdots+ \frac{30^{3}-29^{3}+28^{3}-27^{3}+\ldots+2^{3}-1^{3}}{15 \times 63}\) is equal to _____________.

Let the sum of an infinite G.P., whose first term is a and the common ratio is r, be 5 . Let the sum of its first five terms be \(\frac{98}{25}\). Then the sum of the first 21 terms of an AP, whose first term is $$10\mathrm{a r}, \mathrm{n}...

If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to ___________.

If \(A = \sum\limits_{n = 1}^\infty {{1 \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}}\) and \(B = \sum\limits_{n = 1}^\infty {{{{{( - 1)}^n}} \over {{{\left( {3 + {{( - 1)}^n}} \right)}^n}}}}\), then \({A \over B}\) is equal to :

The series of positive multiples of 3 is divided into sets : \(\{3\},\{6,9,12\},\{15,18,21,24,27\}, \ldots\) Then the sum of the elements in the \(11^{\text {th }}\) set is equal to ____________.

Consider two G.Ps. 2, 22, 23, ..... and 4, 42, 43, .... of 60 and n terms respectively. If the geometric mean of all the 60 + n terms is \({(2)^{{{225} \over 8}}}\), then \(\sum\limits_{k = 1}^n {k(n - k)}\) is equal to :

Different A.P.'s are constructed with the first term 100, the last term 199, and integral common differences. The sum of the common differences of all such A.P.'s having at least 3 terms and at most 33 terms is ___________.

If \(\sum\limits_{k=1}^{10} \frac{k}{k^{4}+k^{2}+1}=\frac{m}{n}\), where m and n are co-prime, then \(m+n\) is equal to _____________.

The greatest integer less than or equal to the sum of first 100 terms of the sequence \({1 \over 3},{5 \over 9},{{19} \over {27}},{{65} \over {81}},\) ...... is equal to ___________.

For a natural number n, let \({\alpha _n} = {19^n} - {12^n}\). Then, the value of \({{31{\alpha _9} - {\alpha _{10}}} \over {57{\alpha _8}}}\) is ___________.

The sum 1 + 2 . 3 + 3 . 32 + ......... + 10 . 39 is equal to :

Let \(a_{1}=b_{1}=1, a_{n}=a_{n-1}+2\) and \(b_{n}=a_{n}+b_{n-1}\) for every natural number \(n \geqslant 2\). Then \(\sum\limits_{n = 1}^{15} {{a_n}.{b_n}}\) is equal to ___________.

Let \(a, b\) be two non-zero real numbers. If \(p\) and \(r\) are the roots of the equation \(x^{2}-8 \mathrm{a} x+2 \mathrm{a}=0\) and \(\mathrm{q}\) and s are the roots of the equation \(x^{2}+12 \mathrm{~b} x+6 \mathrm{~b}=0\), such that...

The sum \(\sum\limits_{n = 1}^{21} {{3 \over {(4n - 1)(4n + 3)}}}\) is equal to

If \(\{ {a_i}\} _{i = 1}^n\), where n is an even integer, is an arithmetic progression with common difference 1, and \(\sum\limits_{i = 1}^n {{a_i} = 192} ,\,\sum\limits_{i = 1}^{n/2} {{a_{2i}} = 120}\), then n is equal to :

Let x, y > 0. If x3y2 = 215, then the least value of 3x + 2y is

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r2, then r2 \(-\) d...

The sum of 10 terms of the series
\({3 \over {{1^2} \times {2^2}}} + {5 \over {{2^2} \times {3^2}}} + {7 \over {{3^2} \times {4^2}}} + ....\) is :

If \(S = {7 \over 5} + {9 \over {{5^2}}} + {{13} \over {{5^3}}} + {{19} \over {{5^4}}} + ....\), then 160 S is equal to ________.

The number of 4-digit numbers which are neither multiple of 7 nor multiple of 3 is ____________.

Let a1, a2, a3, ..... be an A.P. If \({{{a_1} + {a_2} + .... + {a_{10}}} \over {{a_1} + {a_2} + .... + {a_p}}} = {{100} \over {{p^2}}}\), p \(\ne\) 10, then \({{{a_{11}}} \over {{a_{10}}}}\) is equal to :

If \({\log _3}2,{\log _3}({2^x} - 5),{\log _3}\left( {{2^x} - {7 \over 2}} \right)\) are in an arithmetic progression, then the value of x is equal to _____________.

If for x, y \(\in\) R, x > 0, y = log10x + log10x1/3 + log10x1/9 + ...... upto \(\infty\) terms and \({{2 + 4 + 6 + .... + 2y} \over {3 + 6 + 9 + ..... + 3y}} = {4 \over {{{\log }_{10}}x}}\), then the ordered pair (x, y) is equal to :

If 0 < x < 1, then \({3 \over 2}{x^2} + {5 \over 3}{x^3} + {7 \over 4}{x^4} + .....\), is equal to :

If 0 < x < 1 and \(y = {1 \over 2}{x^2} + {2 \over 3}{x^3} + {3 \over 4}{x^4} + ....\), then the value of e1 + y at \(x = {1 \over 2}\) is :

The sum of the infinite series \(1 + {2 \over 3} + {7 \over {{3^2}}} + {{12} \over {{3^3}}} + {{17} \over {{3^4}}} + {{22} \over {{3^5}}} + ......\) is equal to :

In an increasing geometric series, the sum of the second and the sixth term is \({{25} \over 2}\) and the product of the third and fifth term is 25. Then, the sum of 4th, 6th and 8th terms is equal to :

The sum of the series \(\sum\limits_{n = 1}^\infty {{{{n^2} + 6n + 10} \over {(2n + 1)!}}}\) is equal to :

The total number of 4-digit numbers whose greatest common divisor with 18 is 3, is _________.

If the arithmetic mean and geometric mean of the pth and qth terms of the sequence \(-\)16, 8, \(-\)4, 2, ...... satisfy the equation 4x2 \(-\) 9x + 5 = 0, then p + q is equal to __________.

If the sum of an infinite GP a, ar, ar2, ar3, ....... is 15 and the sum of the squares of its each term is 150, then the sum of ar2, ar4, ar6, ....... is :

The sum of the series \({1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}\) when x = 2 is :

Let a1, a2, ......., a10 be an AP with common difference \(-\) 3 and b1, b2, ........., b10 be a GP with common ratio 2. Let ck = ak + bk, k = 1, 2, ......, 10. If c2 = 12 and c3 = 13, then \(\sum\limits_{k = 1}^{10} {{c_k}}\) is equal to ...

The sum of all 3-digit numbers less than or equal to 500, that are formed without using the digit "1" and they all are multiple of 11, is _____________.

If the value of \({\left( {1 + {2 \over 3} + {6 \over {{3^2}}} + {{10} \over {{3^3}}} + ....upto\,\infty } \right)^{{{\log }_{(0.25)}}\left( {{1 \over 3} + {1 \over {{3^2}}} + {1 \over {{3^3}}} + ....upto\,\infty } \right)}}\) is \(l\), the...

Let Sn be the sum of the first n terms of an arithmetic progression. If S3n = 3S2n, then the value of \({{{S_{4n}}} \over {{S_{2n}}}}\) is :

If \(0 < \theta ,\phi < {\pi \over 2},x = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta } ,y = \sum\limits_{n = 0}^\infty {{{\sin }^{2n}}\phi }\) and \(z = \sum\limits_{n = 0}^\infty {{{\cos }^{2n}}\theta .{{\sin }^{2n}}\phi }\) t...

Let A1, A2, A3, ....... be squares such that for each n \(\ge\) 1, the length of the side of An equals the length of diagonal of An+1. If the length of A1 is 12 cm, then the smallest value of n for which area of An is less than one, is __...

The minimum value of \(f(x) = {a^{{a^x}}} + {a^{1 - {a^x}}}\), where a, \(x \in R\) and a > 0, is equal to :

The sum of first four terms of a geometric progression (G. P.) is \({{65} \over {12}}\) and the sum of their respective reciprocals is \({{65} \over {18}}\). If the product of first three terms of the G.P. is 1, and the third term is $$\alp...

The sum of all the elements in the set {n\(\in\) {1, 2, ....., 100} | H.C.F. of n and 2040 is 1} is equal to _____________.

Let Sn denote the sum of first n-terms of an arithmetic progression. If S10 = 530, S5 = 140, then S20 \(-\) S6 is equal to:

Let \(\left\{ {{a_n}} \right\}_{n = 1}^\infty\) be a sequence such that a1 = 1, a2 = 1 and \({a_{n + 2}} = 2{a_{n + 1}} + {a_n}\) for all n \(\ge\) 1. Then the value of \(47\sum\limits_{n = 1}^\infty {{{{a_n}} \over {{2^{3n}}}}}\) is equ...

For k \(\in\) N, let \({1 \over {\alpha (\alpha + 1)(\alpha + 2).........(\alpha + 20)}} = \sum\limits_{K = 0}^{20} {{{{A_k}} \over {\alpha + k}}}\), where \(\alpha > 0\). Then the value of $$100{\left( {{{{A_{14}} + {A_{15}}} \over {...

If sum of the first 21 terms of the series \({\log _{{9^{1/2}}}}x + {\log _{{9^{1/3}}}}x + {\log _{{9^{1/4}}}}x + .......\), where x > 0 is 504, then x is equal to

Let a1, a2, ..........., a21 be an AP such that \(\sum\limits_{n = 1}^{20} {{1 \over {{a_n}{a_{n + 1}}}} = {4 \over 9}}\). If the sum of this AP is 189, then a6a16 is equal to :