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

Three numbers are in an increasing geometric progression with common ration \( r \). 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 \( 3r^2 \), then \( r^2 - d \) is equal to:

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

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

Let \( S_n = 1 \cdot (n - 1) + 2 \cdot (n - 2) + 3 \cdot (n - 3) + \ldots + (n - 1) \cdot 1, \; n \ge 4. \) The sum \( \displaystyle \sum_{n=4}^{\infty}\left( \frac{2 S_n}{n!} - \frac{1}{(n-2)!} \right) \) is equal to:

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

Let \( \dfrac{1}{16}, \, a \) and \( b \) be in G.P. and \( \dfrac{1}{a}, \, \dfrac{1}{b}, \, 6 \) be in A.P., where \( a, b > 0 \). Then \( 72(a+b) \) is equal to ________.

The sum of first four terms of a geometric progression (G.P.) is \( \dfrac{65}{12} \) and the sum of their respective reciprocals is \( \dfrac{65}{18} \). If the product of first three terms of the G.P. is \( 1 \), and the third term is \( \alpha \), then \( 2\alpha \) is ______________.

For \( k\in\mathbb{N} \), let \( \dfrac{1}{\alpha(\alpha+1)(\alpha+2)\ldots(\alpha+20)}=\displaystyle\sum_{k=0}^{20}\dfrac{A_{k}}{\alpha+k} \), where \( \alpha>0 \). Then the value of \( 100\left(\dfrac{A_{14}+A_{15}}{A_{13}}\right)^{2} \) is equal to ________.

Let \( S_n \) be the sum of the first \( n \) terms of an arithmetic progression. If \( S_{3n}=3S_{2n} \) then the value of \( \dfrac{S_{4n}}{S_{2n}} \) is:

Let \( a, b, c \) be in arithmetic progression. Let the centroid of the triangle with vertices \( (a, c) \), \( (2, b) \) and \( (a, b) \) be \( \left( \frac{10}{3}, \frac{7}{3} \right) \). If \( \alpha, \beta \) are the roots of the equation \( a x^2 + b x + 1 = 0 \), then the value of \( \alpha^2 + \beta^2 - \alpha \beta \) is:

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 __________.

Let \(\displaystyle S_n(x) = \log_{a^{1/2}} x + \log_{a^{1/3}} x + \log_{a^{1/6}} x + \log_{a^{1/11}} x + \log_{a^{1/18}} x + \log_{a^{1/27}} x + \dots\dots \) upto \( n \)-terms, where \( a>1 \). If \( S_{24}(x)=1093 \) and \( S_{12}(2x)=265 \), then value of \( a \) is equal to __________.

If \( \alpha, \beta \) are natural numbers such that \( 100^\alpha - 199^\beta = (100)(100) + (99)(101) + (98)(102) + \ldots + (1)(199) \), then the slope of the line passing through \( (\alpha, \beta) \) and origin is:

Let \( a_{1}, a_{2}, \ldots, a_{21} \) be an AP such that \( \displaystyle \sum_{n=1}^{20} \frac{1}{a_{n} a_{n+1}} = \frac{4}{9} \). If the sum of this AP is \( 189 \), then \( a_{6} \cdot a_{16} \) is equal to:

Let \( a_1, a_2, a_3, \ldots \) be an A.P. If \( \dfrac{a_1+a_2+\ldots+a_{10}}{a_1+a_2+\ldots+a_p}=\dfrac{100}{p^2},\; p\neq 10 \), then \( \dfrac{a_{11}}{a_{10}} \) is equal to:

Let \( a_{1},a_{2},\ldots,a_{10} \) be an AP with common difference \( -3 \) and \( b_{1},b_{2},\ldots,b_{10} \) be a GP with common ratio \( 2 \). Let \( c_{k}=a_{k}+b_{k},\, k=1,2,\ldots,10 \). If \( c_{2}=12 \) and \( c_{3}=13 \), then \( \displaystyle \sum_{k=1}^{10}c_{k} \) is equal to __________.

Let \( \{a_n\}_{n=1}^{\infty} \) be a sequence such that \( a_1=1,\, a_2=1 \) and \( a_{n+2}=2a_{n+1}+a_n \) for all \( n\ge 1 \). Then the value of \( 47\displaystyle\sum_{n=1}^{\infty}\frac{a_n}{2^{3n}} \) is equal to ____________.

If \( \log_3 2, \ \log_3(2^x-5), \ \log_3\!\left(2^x-\dfrac{7}{2}\right) \) are in an arithmetic progression, then the value of \( x \) is equal to \( \underline{\quad} \).

If \( [x] \) be the greatest integer less than or equal to \( x \), then \( \displaystyle\sum_{n=8}^{100} \left[ \frac{(-1)^{n} \, n}{2} \right] \) is equal to:

Let \( S_{1} \) be the sum of first \( 2n \) terms of an arithmetic progression. Let \( S_{2} \) be the sum of first \( 4n \) terms of the same arithmetic progression. If \( (S_{2} - S_{1} )\) is \( 1000 \), then the sum of the first \( 6n \) terms of the arithmetic progression is equal to:

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

$\displaystyle \lim_{x \to 2} \left( \displaystyle \sum_{n=1}^{9} \frac{x}{n (n + 1)x^2 + 2 (2n + 1)x + 4} \right) \text{ is equal to:}$

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

If the value of \( \left(1+\dfrac{2}{3}+\dfrac{6}{3^{2}}+\dfrac{10}{3^{3}}+\ldots \text{ up to } \infty \right)^{\log_{(0.25)}\!\left(\dfrac{1}{3}+\dfrac{1}{3^{2}}+\dfrac{1}{3^{3}}+\ldots \text{ up to } \infty \right)} \) is \( l \) then \( l^{2} \) is equal to __________ .

If \( ^{20}C_{r} \) is the co-efficient of \( x^{r} \) in the expansion of \( (1+x)^{20} \), then the value of \( \displaystyle \sum_{r=0}^{20} r^{2}\,^{20}C_{r} \) is equal to:

\( \dfrac{1}{3^2 - 1} + \dfrac{1}{5^2 - 1} + \dfrac{1}{7^2 - 1} + \ldots + \dfrac{1}{(201)^2 - 1} \) is equal to:

Let \( S_n \) denote the sum of first \( n \)-terms of an arithmetic progression. If \( S_{10}=530, S_{5}=140 \), then \( S_{20}-S_{6} \) is equal to:

If \( {}^{1}P_{1}+2\cdot {}^{2}P_{2}+3\cdot {}^{3}P_{3}+\ldots+15\cdot {}^{15}P_{15}={}^qP_{r-s},\; 0\le s\le 1 \), then \( {}^{\,q+s}C_{\,r-s} \) is equal to ________.

Let \( a, b \) and \( c \) be distinct positive numbers. If the vectors \( a\hat{i} + a\hat{j} + c\hat{k} \), \( \hat{i} + \hat{k} \) and \( c\hat{i} + c\hat{j} + b\hat{k} \) are co-planar, then \( c \) is equal to:

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

If the sum of an infinite GP \( a, ar, ar^{2}, ar^{3}, \ldots \) is \( 15 \) and the sum of the squares of its each term is \( 150 \), then the sum of \( ar^{2}, ar^{4}, ar^{6}, \ldots \) is:

If for \( x, y \in \mathbb{R} , \ x > 0 \), \( y = \log_{10} x + \log_{10} x^{1/3} + \log_{10} x^{1/9} + \ldots \) upto \( \infty \) terms and \( \dfrac{2 + 4 + 6 + \ldots + 2y}{3 + 6 + 9 + \ldots + 3y} = \dfrac{4}{\log_{10} x} \), then the ordered pair \( (x, y) \) is equal to:

The mean of \( 10 \) numbers \( 7 \times 8, 10 \times 10, 13 \times 12, 16 \times 14, \ldots \) is __________ .

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JEE Main 2021 25th July Evening Shift

Let \( a, b \) and \( c \) be distinct positive numbers. If the vectors \( a\hat{i} + a\hat{j} + c\hat{k} \), \( \hat{i} + \hat{k} \) and \( c\hat{i} + c\hat{j} + b\hat{k} \) are co-planar, then \( c \) is equal to: