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The sum \( 1 + 2 \cdot 3 + 3 \cdot 3^{2} + \ldots + 10 \cdot 3^{9} \) is equal to:

If \( \dfrac{1}{(20-a)(40-a)}+\dfrac{1}{(40-a)(60-a)}+\ldots+\dfrac{1}{(180-a)(200-a)}=\dfrac{1}{256} \), then the maximum value of \( a \) is:

If the minimum value of \( f(x)=\dfrac{5x^2}{2}+\dfrac{\alpha}{x^5}, \; x>0 \), is \( 14 \), then the value of \( \alpha \) is equal to:

Let \( a_1,a_2,a_3,\ldots \) be an A.P. If \(\displaystyle \sum_{r=1}^{\infty}\frac{a_r}{2^r}=4 \), then \( 4a_2 \) is equal to __________.

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

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

The sum of all the elements of the set \( \{\alpha\in\{1,2,\ldots,100\}\!:\ \mathrm{HCF}(\alpha,24)=1\} \) is ________.

The sum \( \displaystyle\sum_{n=1}^{21}\frac{3}{(4n-1)(4n+3)} \) is equal to:

Let \( C_r \) denote the binomial coefficient of \( x^r \) in the expansion of \( (1+x)^{10} \). If for \( \alpha, \beta \in \mathbb{R} \)
\( C_1 + 3\cdot 2\, C_2 + 5\cdot 3\, C_3 + \ldots \text{ up to } 10 \text{ terms} = \dfrac{\alpha \times 2^{11}}{2^{\beta}-1}\left( C_0 + \dfrac{C_1}{2} + \dfrac{C_2}{3} + \ldots \text{ up to } 10 \text{ terms} \right) \)
then value of \( \alpha + \beta \) is equal to ______.

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

Suppose \( a_{1},a_{2},\ldots ,a_{n},\ldots \) 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 sum of the first ten terms of the progression is equal to:

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

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

Let \( \alpha, \beta \) and \( \gamma \) be three positive real numbers. Let \( f(x)=\alpha x^5+\beta x^3+\gamma x, \, x\in \mathbb{R} \) and \( g:\mathbb{R}\to\mathbb{R} \) be such that \( g\big(f(x)\big)=x \) for all \( x\in \mathbb{R} \). If \( a_1,a_2,a_3,\ldots,a_n \) be in arithmetic progression with mean zero, then the value of \( f\!\left(g\!\left(\dfrac{1}{n}\displaystyle\sum_{r=1}^{n} f(a_i)\right)\right) \) is equal to:

If \( a_1(>0), a_2, a_3, a_4, a_5 \) are in a G.P., \( a_2 + a_4 = 2a_3 + 1 \) and \( 3a_2 + a_3 = 2a_4 \), then \( a_2 + a_4 + 2a_5 \) is equal to _______.

If \( \frac{1}{2\cdot 3^{10}} + \frac{1}{2^{2}\cdot 3^{9}} + \ldots + \frac{1}{2^{10}\cdot 3} = \frac{K}{2^{10}\cdot 3^{10}} \) then the remainder when \( K \) is divded by \( 6 \) is:

Let \( A = \{1,a_1,a_2,\ldots,a_{18},77\} \) be a set of integers with \( 1 < a_1 < a_2 < \ldots < a_{18} < 77 \).
Let the set \( A + A = \{ x + y : x, y \in A \} \) contain exactly \( 39 \) elements. Then, the value of \( a_1 + a_2 + \ldots + a_{18} \) is equal to ________.

Consider two G. Ps. \( 2,\,2^{2},\,2^{3},\ldots \) and \( 4,\,4^{2},\,4^{3},\ldots \) of \( 60 \) and \( n \) terms respectively. If the geometric mean of all the \( 60+n \) terms is \( (2)^{\dfrac{225}{8}} \), then \( \displaystyle \sum_{k=1}^{n} k\,(n-k) \) is equal to:

\(\displaystyle \sum_{r=1}^{20} (r^2 + 1)(r!) \) is equal to

Let \( A_1, A_2, A_3, \ldots \) be an increasing geometric progression of positive real numbers. If \( A_1 A_3 A_5 A_7 = \frac{1}{1296} \) and \( A_2 + A_4 = \frac{7}{36} \), then, the value of \( A_6 + A_8 + A_{10} \) is equal to:

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

The total number of \( 3 \)-digit numbers, whose greatest common divisor with \( 36 \) is \( 2 \), is _______.

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

If \(\displaystyle \frac{6}{3^{12}} + \frac{10}{3^{11}} + \frac{20}{3^{10}} + \frac{40}{3^{9}} + \ldots + \frac{10240}{3} = 2^{n} \cdot m \) , where \( m \) is odd, then \( m \cdot n \) is equal to _______________.

Let \( x, y > 0 \). If \( x^3 y^2 = 2^{15} \), then the least value of \( 3x + 2y \) is:

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

Let \( a_1 = b_1 = 1, \ a_n = a_{n-1} + 2 \ \& \ b_n = a_n + b_{n-1} \) for every natural number \( n \ge 2 \). Then
\( \displaystyle\sum_{n=1}^{15} a_n \cdot b_n \) is equal to ________.

Consider the sequence \( a_1, a_2, a_3, \ldots \) such that \( a_1 = 1, a_2 = 2 \) and \( a_{n+2} = \dfrac{2}{a_{n+1}} + a_n \) for \( n = 1, 2, 3, \ldots \). If
\( \left( \dfrac{a_1 + \dfrac{1}{a_2}}{a_3} \right)\left( \dfrac{a_2 + \dfrac{1}{a_3}}{a_4} \right)\left( \dfrac{a_3 + \dfrac{1}{a_4}}{a_5} \right) \ldots \left( \dfrac{a_{30} + \dfrac{1}{a_{31}}}{a_{32}} \right) = 2^{\alpha}\left( {}^{61}C_{31} \right) \) , then \( \alpha \) is equal to:

If \( a_1, a_2, a_3, \ldots \) and \( b_1, b_2, b_3, \ldots \) are A.P., and \( a_1 = 2, \; a_{10} = 3, \; a_1 b_1 = 1 = a_{10} b_{10} \), then \( a_4 b_4 \) is equal to

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

If \( A = \displaystyle \sum_{n=1}^{\infty} \frac{1}{\left(3 + (-1)^n\right)^n} \) and \( B = \displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n}{\left(3 + (-1)^n\right)^n} \) then \( \frac{A}{B} \) is equal to:

Let \( A = \displaystyle \sum_{i=1}^{10}\ \displaystyle \sum_{j=1}^{10} \min\{i, j\} \) and \( B =\displaystyle \sum_{i=1}^{10}\ \displaystyle \sum_{j=1}^{10} \max\{i, j\} \). Then \( A + B \) is equal to ____________ .

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

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

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

Let \( A = \{ n \in \mathbb{N} : H.C.F.(n, 45) = 1 \} \) and Let \( B = \{ 2k : k \in \{ 1, 2, \ldots , 100 \} \} \). Then the sum of all the elements of \( A \cap B \) is ____________ .

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JEE Main 2022 25th June Morning Shift

If \( \frac{1}{2\cdot 3^{10}} + \frac{1}{2^{2}\cdot 3^{9}} + \ldots + \frac{1}{2^{10}\cdot 3} = \frac{K}{2^{10}\cdot 3^{10}} \) then the remainder when \( K \) is divded by \( 6 \) is: