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Let \( 3,\,7,\,11,\,15,\,\ldots,\,403 \) and \( 2,\,5,\,8,\,11,\,\ldots,\,404 \) be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ______.

Let the positive integers be written in the form:

If the \( k^{\text{th}} \) row contains exactly \( k \) numbers for every natural number \( k \), then the row in which the number \( 5310 \) will be, is _______.

Let \( S_n \) denote the sum of the first \( n \) terms of an arithmetic progression. If \( S_{10} = 390 \) and the ratio of the tenth and the fifth terms is \( 15 : 7 \), then \( S_{15} - S_{5} \) is equal to:

Let \( S_{n} \) be the sum to \( n \)-terms of an arithmetic progression \( 3, \, 7, \, 11, \, \ldots \). If \( 40 < \left( \dfrac{6}{n(n+1)} \displaystyle\sum_{k=1}^{n} S_{k} \right) < 42 \), then \( n \) equals ______.

The value of \( \dfrac{1\times 2^2 + 2\times 3^2 + \dots + 100\times (101)^2}{1^2\times 2 + 2^2\times 3 + \dots + 100^2\times 101} \) is:

Let the first three terms \( 2 \), \( p \) and \( q \), with \( q \neq 2 \), of a G.P. be respectively the \( 7^{\text{th}} \), \( 8^{\text{th}} \) and \( 13^{\text{th}} \) terms of an A.P. If the \( 5^{\text{th}} \) term of the G.P. is the \( n^{\text{th}} \) term of the A.P., then \( n \) is equal to:

If each term of a geometric progression \( a_{1},a_{2},a_{3},\ldots \) with \( a_{1}=\dfrac{1}{8} \) and \( a_{2}\neq a_{1} \), is the arithmetic mean of the next two terms and \( S_{n}=a_{1}+a_{2}+\cdots +a_{n} \), then \( S_{20}-S_{18} \) is equal to:

If \( 2\tan^2\theta-5\sec\theta-1 \) has exactly \( 7 \) solutions in the interval \(\displaystyle \left[0,\frac{n\pi}{2}\right] \) for the least value of \( n\in \mathbb{N} \), then \(\displaystyle \sum_{k=1}^{n} \frac{k}{2^{k}} \) is equal to:

The sum of the series \( \frac{1}{1-3\cdot 1^{2}+1^{4}} + \frac{2}{1-3\cdot 2^{2}+2^{4}} + \frac{3}{1-3\cdot 3^{2}+3^{4}} + \ldots \) up to \( 10 \) terms is:

For \( x \ge 0 \) the least value of \( K \), for which \( 4^{1+x}+4^{1-x},\; \dfrac{K}{2},\; 16^{x}+16^{-x} \) are three consecutive terms of an A.P., is equal to:

Let \( S_n \) denote the sum of first \( n \) terms of an arithmetic progression. If \( S_{20} = 790 \) and \( S_{10} = 145 \), then \( S_{15} - S_{5} \) is:

If \( 1 + \frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}} + \frac{5-2\sqrt{6}}{18} + \frac{9\sqrt{3}-11\sqrt{2}}{36\sqrt{3}} + \frac{49-20\sqrt{6}}{180} + \ldots \) upto \( \infty = 2 + \left( \sqrt{\frac{b}{a}} + 1 \right) \log_{e}\!\left( \frac{a}{b} \right) \), where \( a \) and \( b \) are integers with \( \gcd(a,b)=1 \), then \( 11a + 18b \) is equal to \( \_\_\_\_\_\_\_\_ \).

If \( \log_{e}a, \log_{e}b, \log_{e}c \) are in an A.P. and \( \log_{e}a-\log_{e}2b, \log_{e}2b-\log_{e}3c, \log_{e}3c-\log_{e}a \) are also in an A.P., then \( a:b:c \) is equal to:

If three successive terms of G.P. with common ratio \( r\,(r>1) \) are the lengths of the sides of a triangle and \( [r] \) denotes the greatest integer less than or equal to \( r \), then \( 3[r]+[-r] \) is equal to __________.

If \( \frac{1}{\sqrt{1}+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\ldots+\frac{1}{\sqrt{99}+\sqrt{100}} = m \) and \( \frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\ldots+\frac{1}{99\cdot 100} = n \) then the point \( (m,n) \) lies on the line:

Let $A = \{n \in [100, 700] \cap \mathbb{N} : n \text{ is neither a multiple of } 3 \text{ nor a multiple of } 4\}$. Then the number of elements in $A$ is:

If the sum of the series \( \frac{1}{1 \cdot (1 + d)} + \frac{1}{(1 + d)(1 + 2d)} + \ldots + \frac{1}{(1 + 9d)(1 + 10d)} \) is equal to \( 5 \), then \( 50d \) is equal to:

An arithmetic progression is written in the following way
The sum of all the terms of the \( 10^{\text{th}} \) row is __________.

Let \( a \) and \( b \) be two distinct positive real numbers. Let \( 11^{\text{th}} \) term of a GP, whose first term is \( a \) and third term is \( b \), is equal to \( p^{\text{th}} \) term of another GP, whose first term is \( a \) and fifth term is \( b \). Then \( p \) is equal to:

Let \( 3, a, b, c \) be in A.P. and \( 3, a - 1, b + 1, c + 9 \) be in G.P. Then, the arithmetic mean of \( a, b \) and \( c \) is:

Let \( a_{1}, a_{2}, a_{3}, \ldots \) be in an arithmetic progression of positive terms.
Let \( A_{k} = a_{1}^{2} - a_{2}^{2} + a_{3}^{2} - a_{4}^{2} + \cdots + a_{2k-1}^{2} - a_{2k}^{2} \) If \( A_{3} = -153 \), \( A_{5} = -435 \) and \( a_{1}^{2} + a_{2}^{2} + a_{3}^{2} = 66 \), then \( a_{17} - A_{7} \) is equal to ____________.

Let the first term of a series be \( T_1 = 6 \) and its \( r^{\text{th}} \) term \( T_r = 3T_{r-1} + 6^r, \; r = 2, 3, \ldots, n \). If the sum of the first \( n \) terms of this series is \( \frac{1}{5}\left(n^2 - 12n + 39\right)\!\left(4\cdot 6^n - 5\cdot 3^n + 1\right) \), then \( n \) is equal to ________.

Let \( 2^{\text{nd}},8^{\text{th}} \) and \( 44^{\text{th}} \) terms of a non-constant A.P. be respectively the \( 1^{\text{st}},2^{\text{nd}} \) and \( 3^{\text{rd}} \) terms of a G.P.. If the first term of the A.P. is \( 1 \), then the sum of its first \( 20 \) terms is equal to:

Let \( \alpha = 1^2 + 4^2 + 8^2 + 13^2 + 19^2 + 26^2 + \ldots \) up to \( 10 \) terms and \( \beta = \displaystyle \sum_{n=1}^{10} n^4 \). If \( 4\alpha - \beta = 55k + 40 \), then \( k \) is equal to ________.

If \( 8 = 3 + \frac{1}{4}(3 + p) + \frac{1}{4^{2}}(3 + 2p) + \frac{1}{4^{3}}(3 + 3p) + \ldots , \; \infty \) , then the value of \( p \) is ________.

The number of common terms in the progressions \( 4, 9, 14, 19, \ldots \), up to \( 25^{\text{th}} \) term and \( 3, 6, 9, 12, \ldots \), up to \( 37^{\text{th}} \) term is:

The \( 20^{\text{th}} \) term from the end of the progression \(\displaystyle 20,\, 19\frac{1}{4},\, 18\frac{1}{2},\, 17\frac{3}{4},\, \ldots ,\, -129\frac{1}{4} \) is:

A software company sets up \( m \) number of computer systems to finish an assignment in \( 17 \) days. If \( 4 \) computer systems crashed on the start of the second day, \( 4 \) more computer systems crashed on the start of the third day and so on, then it took \( 8 \) more days to finish the assignment. The value of \( m \) is equal to:

If \( S(x)=(1+x)+2(1+x)^{2}+3(1+x)^{3}+\ldots+60(1+x)^{60},\; x\neq 0 \), where \( a,\; b\in \mathbb{N} \) then \( (a+b) \) equal to ______.

If in a G.P. of \( 64 \) terms, the sum of all the terms is \( 7 \) times the sum of the odd terms of the G.P., then the common ratio of the G.P. is equal to:

Let \( a, ar, ar^{2},\ldots \) be an infinite G.P. If \( \sum\limits_{n=0}^{\infty}ar^{n}=57 \) and \( \sum\limits_{n=0}^{\infty}a^{3}r^{3n}=9747 \), then \( a+18r \) is equal to

Let three real numbers \( a, b, c \) be in arithmetic progression and \( a + 1, b, c + 3 \) be in geometric progression. If \( a > 10 \) and the arithmetic mean of \( a, b \) and \( c \) is \( 8 \), then the cube of the geometric mean of \( a, b \) and \( c \) is:

If \(\displaystyle \left( \frac{1}{\alpha+1} + \frac{1}{\alpha+2} + \cdots + \frac{1}{\alpha+1012} \right) - \left( \frac{1}{2\cdot 1} + \frac{1}{4\cdot 3} + \frac{1}{6\cdot 5} + \cdots + \frac{1}{2024\cdot 2023} \right) = \frac{1}{2024} \), then \( \alpha \) is equal to _________.

In an increasing geometric progression of positive terms, the sum of the second and sixth terms is \( \frac{70}{3} \) and the product of the third and fifth terms is \( 49 \). Then the sum of the \( 4^{\text{th}} \), \( 6^{\text{th}} \) and \( 8^{\text{th}} \) terms is equal to:

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JEE Main 2024 1st Feb Morning Shift

Let \( 3,\,7,\,11,\,15,\,\ldots,\,403 \) and \( 2,\,5,\,8,\,11,\,\ldots,\,404 \) be two arithmetic progressions. Then the sum, of the common terms in them, is equal to ______.