Sequences and Series

If
\((20)^{19}+2(21)(20)^{18}+3(21)^{2}(20)^{17}+\ldots+20(21)^{19}=k(20)^{19}\),
then \(k\) is equal to ___________.

If \(\operatorname{gcd}~(\mathrm{m}, \mathrm{n})=1\) and \(1^{2}-2^{2}+3^{2}-4^{2}+\ldots . .+(2021)^{2}-(2022)^{2}+(2023)^{2}=1012 ~m^{2} n\) then \(m^{2}-n^{2}\) is equal to :

Let \(a_{1}, a_{2}, \ldots, a_{n}\) be in A.P. If \(a_{5}=2 a_{7}\) and \(a_{11}=18\), then
\(12\left(\frac{1}{\sqrt{a_{10}}+\sqrt{a_{11}}}+\frac{1}{\sqrt{a_{11}}+\sqrt{a_{12}}}+\ldots+\frac{1}{\sqrt{a_{17}}+\sqrt{a_{18}}}\right)\) is equa...

If the sum and product of four positive consecutive terms of a G.P., are 126 and 1296 , respectively, then the sum of common ratios of all such GPs is

The sum $1^{2}-2 \cdot 3^{2}+3 \cdot 5^{2}-4 \cdot 7^{2}+5 \cdot 9^{2}-\ldots+15 \cdot 29^{2}$ is _________.

Let $a_1, a_2, a_3, \ldots$ be an A.P. If $a_7=3$, the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero, then $n !-4 a_{n(n+2)}$ is equal to :

Let \(\sum_\limits{n=0}^{\infty} \frac{\mathrm{n}^{3}((2 \mathrm{n}) !)+(2 \mathrm{n}-1)(\mathrm{n} !)}{(\mathrm{n} !)((2 \mathrm{n}) !)}=\mathrm{ae}+\frac{\mathrm{b}}{\mathrm{e}}+\mathrm{c}\), where $$\mathrm{a}, \mathrm{b}, \mathrm{c} \in...

If \({a_n} = {{ - 2} \over {4{n^2} - 16n + 15}}\), then \({a_1} + {a_2}\, + \,....\, + \,{a_{25}}\) is equal to :

The $8^{\text {th }}$ common term of the series
$$ \begin{aligned} & S_1=3+7+11+15+19+\ldots . . \\\\ & S_2=1+6+11+16+21+\ldots . . \end{aligned} $$
is :

Let $a, b, c>1, a^3, b^3$ and $c^3$ be in A.P., and $\log _a b, \log _c a$ and $\log _b c$ be in G.P. If the sum of first 20 terms of an A.P., whose first term is $\frac{a+4 b+c}{3}$ and the common difference is $\frac{a-8 b+c}{10}$ is $-44...

Let \(a_1,a_2,a_3,...\) be a \(GP\) of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then \(a_1a_9+a_2a_4a_9+a_5+a_7\) is equal to __________.

Let \(\{ {a_k}\}\) and \(\{ {b_k}\} ,k \in N\), be two G.P.s with common ratios \({r_1}\) and \({r_2}\) respectively such that \({a_1} = {b_1} = 4\) and \({r_1} < {r_2}\). Let \({c_k} = {a_k} + {b_k},k \in N\). If \({c_2} = 5\) and $${c_3}...

Let \(a_1=b_1=1\) and \({a_n} = {a_{n - 1}} + (n - 1),{b_n} = {b_{n - 1}} + {a_{n - 1}},\forall n \ge 2\). If \(S = \sum\limits_{n = 1}^{10} {{{{b_n}} \over {{2^n}}}}\) and \(T = \sum\limits_{n = 1}^8 {{n \over {{2^{n - 1}}}}}\), then $${...

For the two positive numbers \(a,b,\) if \(a,b\) and \(\frac{1}{18}\) are in a geometric progression, while \(\frac{1}{a},10\) and \(\frac{1}{b}\) are in an arithmetic progression, then \(16a+12b\) is equal to _________.

The 4\(^\mathrm{th}\) term of GP is 500 and its common ratio is \(\frac{1}{m},m\in\mathbb{N}\). Let \(\mathrm{S_n}\) denote the sum of the first n terms of this GP. If \(\mathrm{S_6 > S_5 + 1}\) and \(\mathrm{S_7 < S_6 + \frac{1}{2}}\), the...

For three positive integers p, q, r, \({x^{p{q^2}}} = {y^{qr}} = {z^{{p^2}r}}\) and r = pq + 1 such that 3, 3 log\(_yx\), 3 log\(_zy\), 7 log\(_xz\) are in A.P. with common difference \(\frac{1}{2}\). Then r-p-q is equal to

If \({{{1^3} + {2^3} + {3^3}\, + \,...\,up\,to\,n\,terms} \over {1\,.\,3 + 2\,.\,5 + 3\,.\,7\, + \,...\,up\,to\,n\,terms}} = {9 \over 5}\), then the value of \(n\) is

Let \(a_{1}=8, a_{2}, a_{3}, \ldots, a_{n}\) be an A.P. If the sum of its first four terms is 50 and the sum of its last four terms is 170 , then the product of its middle two terms is ___________.

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

The sum of the common terms of the following three arithmetic progressions.
\(3,7,11,15, \ldots ., 399\),
\(2,5,8,11, \ldots ., 359\) and
\(2,7,12,17, \ldots ., 197\),
is equal to _____________.

The sum \(\sum\limits_{n = 1}^\infty {{{2{n^2} + 3n + 4} \over {(2n)!}}}\) is equal to :

If the sum of the series

$\left(\frac{1}{2}-\frac{1}{3}\right)+\left(\frac{1}{2^{2}}-\frac{1}{2 \cdot 3}+\frac{1}{3^{2}}\right)+\left(\frac{1}{2^{3}}-\frac{1}{2^{2} \cdot 3}+\frac{1}{2 \cdot 3^{2}}-\frac{1}{3^{3}}\right)+$

$\left(\frac{1}...

Let $A_{1}$ and $A_{2}$ be two arithmetic means and $G_{1}, G_{2}, G_{3}$ be three geometric means of two distinct positive numbers. Then $G_{1}^{4}+G_{2}^{4}+G_{3}^{4}+G_{1}^{2} G_{3}^{2}$ is equal to :

The sum to \(20\) terms of the series \(2 \cdot 2^{2}-3^{2}+2 \cdot 4^{2}-5^{2}+2 \cdot 6^{2}-\ldots \ldots\) is equal to __________.

Let \(s_{1}, s_{2}, s_{3}, \ldots, s_{10}\) respectively be the sum to 12 terms of 10 A.P. s whose first terms are \(1,2,3, \ldots .10\) and the common differences are \(1,3,5, \ldots \ldots, 19\) respectively. Then $$\sum_\limits{i=1}^{10}...

Let a\(_1\), a\(_2\), a\(_3\), .... be a G.P. of increasing positive numbers. Let the sum of its 6th and 8th terms be 2 and the product of its 3rd and 5th terms be \(\frac{1}{9}\). Then \(6(a_2+a_4)(a_4+a_6)\) is equal to

Let \(< a_{\mathrm{n}} >\) be a sequence such that \(a_{1}+a_{2}+\ldots+a_{n}=\frac{n^{2}+3 n}{(n+1)(n+2)}\). If \(28 \sum_\limits{k=1}^{10} \frac{1}{a_{k}}=p_{1} p_{2} p_{3} \ldots p_{m}\), where $$\mathrm{p}_{1}, \mathrm{p}_{2}, \ldots ....

Let \(S=109+\frac{108}{5}+\frac{107}{5^{2}}+\ldots .+\frac{2}{5^{107}}+\frac{1}{5^{108}}\). Then the value of \(\left(16 S-(25)^{-54}\right)\) is equal to ___________.

Let \(x_{1}, x_{2}, \ldots, x_{100}\) be in an arithmetic progression, with \(x_{1}=2\) and their mean equal to 200 . If \(y_{i}=i\left(x_{i}-i\right), 1 \leq i \leq 100\), then the mean of \(y_{1}, y_{2}, \ldots, y_{100}\) is :

For \(k \in \mathbb{N}\), if the sum of the series \(1+\frac{4}{k}+\frac{8}{k^{2}}+\frac{13}{k^{3}}+\frac{19}{k^{4}}+\ldots\) is 10 , then the value of \(k\) is _________.

Let \(a, b, c\) and \(d\) be positive real numbers such that \(a+b+c+d=11\). If the maximum value of \(a^{5} b^{3} c^{2} d\) is \(3750 \beta\), then the value of \(\beta\) is

The sum of all those terms, of the arithmetic progression 3, 8, 13, ...., 373, which are not divisible by 3, is equal to ____________.

Let the first term \(\alpha\) and the common ratio r of a geometric progression be positive integers. If the sum of squares of its first three terms is 33033, then the sum of these three terms is equal to

Suppose \(a_{1}, a_{2}, 2, a_{3}, a_{4}\) be in an arithmetico-geometric progression. If the common ratio of the corresponding geometric progression is 2 and the sum of all 5 terms of the arithmetico-geometric progression is $$\frac{49}{2}$...

If \(\mathrm{S}_{n}=4+11+21+34+50+\ldots\) to \(n\) terms, then \(\frac{1}{60}\left(\mathrm{~S}_{29}-\mathrm{S}_{9}\right)\) is equal to :

Let for \(f(x) = {a_0}{x^2} + {a_1}x + {a_2},\,f'(0) = 1\) and \(f'(1) = 0\). If a0, a1, a2 are in an arithmatico-geometric progression, whose corresponding A.P. has common difference 1 and corresponding G.P. has common ratio 2, then f(4) i...

The value of \(1 + {1 \over {1 + 2}} + {1 \over {1 + 2 + 3}} + \,\,....\,\, + \,\,{1 \over {1 + 2 + 3 + \,\,.....\,\, + \,\,11}}\) is equal to:

Let \(\{ {a_n}\} _{n = 0}^\infty\) be a sequence such that \({a_0} = {a_1} = 0\) and \({a_{n + 2}} = 2{a_{n + 1}} - {a_n} + 1\) for all n \(\ge\) 0. Then, \(\sum\limits_{n = 2}^\infty {{{{a_n}} \over {{7^n}}}}\) is equal to:

Let 3, 6, 9, 12, ....... upto 78 terms and 5, 9, 13, 17, ...... upto 59 terms be two series. Then, the sum of the terms common to both the series is equal to ________.

The sum of the infinite series \(1 + {5 \over 6} + {{12} \over {{6^2}}} + {{22} \over {{6^3}}} + {{35} \over {{6^4}}} + {{51} \over {{6^5}}} + {{70} \over {{6^6}}} + \,\,.....\) is equal to :

If \(\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\frac{1}{4 \times 5 \times 6}+\ldots+\frac{1}{100 \times 101 \times 102}=\frac{\mathrm{k}}{101}\), then 34 k is equal to _________.

Let \(a_{1}, a_{2}, a_{3}, \ldots\) be an A.P. If \(\sum\limits_{r=1}^{\infty} \frac{a_{r}}{2^{r}}=4\), then \(4 a_{2}\) is equal to _________.

$$ \begin{aligned} &\text { Let }\left\{a_{n}\right\}_{n=0}^{\infty} \text { be a sequence such that } a_{0}=a_{1}=0 \text { and } \\\\ &a_{n+2}=3 a_{n+1}-2 a_{n}+1, \forall n \geq 0 . \end{aligned} $$
Then $$a_{25} a_{23}-2 a_{25} a_{22}-2...

Let A = {1, a1, a2 ....... a18, 77} be a set of integers with 1 < a1 < a2 < ....... < a18 < 77. Let the set A + A = {x + y : x, y \(\in\) A} contain exactly 39 elements. Then, the value of a1 + a2 + ...... + a18 is equal to _____________.

Let A1, A2, A3, ....... be an increasing geometric progression of positive real numbers. If A1A3A5A7 = \({1 \over {1296}}\) and A2 + A4 = \({7 \over {36}}\), then the value of A6 + A8 + A10 is equal to

Let for n = 1, 2, ......, 50, Sn be the sum of the infinite geometric progression whose first term is n2 and whose common ratio is \({1 \over {{{(n + 1)}^2}}}\). Then the value of $${1 \over {26}} + \sum\limits_{n = 1}^{50} {\left( {{S_n} +...

If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is :

Consider the sequence \(a_{1}, a_{2}, a_{3}, \ldots\) such that \(a_{1}=1, a_{2}=2\) and \(a_{n+2}=\frac{2}{a_{n+1}}+a_{n}\) for \(\mathrm{n}=1,2,3, \ldots .\) If $$\left(\frac{\mathrm{a}_{1}+\frac{1}{\mathrm{a}_{2}}}{\mathrm{a}_{3}}\right)...

\({6 \over {{3^{12}}}} + {{10} \over {{3^{11}}}} + {{20} \over {{3^{10}}}} + {{40} \over {{3^9}}} + \,\,...\,\, + \,\,{{10240} \over 3} = {2^n}\,.\,m\), where m is odd, then m . n is equal to ____________.

If the sum of the first ten terms of the series
\({1 \over 5} + {2 \over {65}} + {3 \over {325}} + {4 \over {1025}} + {5 \over {2501}} + \,\,....\)
is \({m \over n}\), where m and n are co-prime numbers, then m + n is equal to _____________...