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Suppose \( a, b, c \) are in A.P. and \( a^{2}, 2b^{2}, c^{2} \) are in G.P. If \( a < b < c \) and \( a + b + c = 1 \), then \( 9(a^{2} + b^{2} + c^{2}) \) is equal to __________ .

The sum \( \frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \ldots \) up to \( 8 \) terms, is:

\( \left( \frac{1}{3} + \frac{4}{7} \right) + \left( \frac{1}{3^2} + \frac{1}{3} \times \frac{4}{7} + \frac{4^2}{7^2} \right) + \left( \frac{1}{3^3} + \frac{1}{3^2} \times \frac{4}{7} + \frac{1}{3} \times \frac{4^2}{7^2} + \frac{4^3}{7^3} \right) + \ldots \) upto infinite terms, is equal to:

If the sum of the first \( 10 \) terms of the series \( \dfrac{1}{1+1^{4}\times 4}+\dfrac{2}{1+2^{4}\times 4}+\dfrac{3}{1+3^{4}\times 4}+\dfrac{4}{1+4^{4}\times 4}+\cdots \) is \( \dfrac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to:

The value of \( \displaystyle \sum_{k=1}^{\infty} (-1)^{k+1} \, \left( \frac{k(k+1)}{k!} \right) \) is:

Let \( a_1, \frac{a_2}{2}, \frac{a_3}{2^2}, \ldots, \frac{a_{10}}{2^9} \) be a G.P. of common ratio \( \frac{1}{\sqrt{2}} \). If \( a_1+a_2+\cdots +a_{10}=62 \), then \( a_1 \) is equal to:

If the sum of the first four terms of an A.P. is \( 6 \) and the sum of its first six terms is \( 4 \), then the sum of its first twelve terms is

For the function \( f(\theta)=\alpha \tan^2\theta+\beta \cot^2\theta \), and \( g(\theta)=\alpha \sin^2\theta+\beta \cos^2\theta, \ \alpha,\beta>0 \), let If the first terms of a G.P. is \( \left(\frac{\alpha}{2\beta}\right) \), its common ratio is \( \left(\frac{2\beta}{\alpha}\right) \) and the sum of its first \( 10 \) terms is \( \frac{m}{n} \), \( \gcd(m,n)=1 \), then \( m+n \) is equal to ______.

Let \( a_1, a_2, a_3, \ldots \) be a G.P. of increasing positive terms such that \( a_2 \cdot a_3 \cdot a_4 = 64 \) and \( a_1 + a_3 + a_5 = \frac{813}{7} \). Then \( a_3 + a_5 + a_7 \) is equal to:

Let \( 729, 81, 9, 1, \ldots \) be a sequence and \( P_n \) denote the product of the first \( n \) terms of this sequence.
If \( 2\sum\limits_{n=1}^{40} (P_n)^{\frac{1}{n}} = \frac{3^{\alpha} - 1}{3^{\beta}} \) and \( \gcd(\alpha, \beta) = 1 \), then \( \alpha + \beta \) is equal to:

The sum of the first ten terms of an A.P. is \( 160 \) and the sum of the first two terms of a G.P. is \( 8 \). If the term of the A.P. is equal to the common ratio of the G.P. and the first term of the G.P. is equal to common difference of the A.P., then the sum of all possible values of the first term of the G.P. is:

The value of \( 1^{3}-2^{3}+3^{3}-\cdots+15^{3} \) is:

\( \displaystyle \sum_{n=1}^{10} \left(\dfrac{528}{n(n+1)(n+2)}\right) \) is equal to:

Let \( \alpha, \beta \) be the roots of the equation \( x^2 - x + p = 0 \) and \( \gamma, \delta \) be the roots the equation \( x^2 - 4x + q = 0; \ p, q \in \mathbb{Z} \). If \( \alpha, \beta, \gamma, \delta \) are in G.P., then \( |p+q| \) equals:

Let \( A \) be the set of first \( 101 \) terms of an A.P., whose first term is \( 1 \) and the common difference is \( 5 \) and let \( B \) be the set of first \( 71 \) terms of an A.P., whose first term is \( 9 \) and the common difference is \( 7 \). Then the number of elements in \( A \cap B \), which are divisible by \( 3 \), is:

Let \( a_1 = 1 \) and for \( n \ge 1 \), \( a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \). Then \( \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \) is equal to _________.

If \( \sum_{k=1}^{n} a_{k}=6n^{3} \), then \( \sum_{k=1}^{6}\left(\frac{a_{k+1}-a_{k}}{36}\right)^{2} \) is equal to _________.

Let \( A_1, A_2, A_3, \ldots, A_{39} \) be \( 39 \) arithmetic means between the numbers \( 59 \) and \( 159 \). Then the mean of \( A_{25}, A_{28}, A_{31} \) and \( A_{36} \) is equal to:

Let \( \alpha = 3 + 4 + 8 + 9 + 13 + 14 + \ldots \) upto \( 40 \) terms. If \( (\tan \beta)^{\frac{\alpha}{1020}} \) is a root of the equation \( x^2 + x - 2 = 0 \), \( \beta \in \left(0, \frac{\pi}{2}\right) \), then \( \sin^2 \beta + 3 \cos^2 \beta \) is equal to:

Let the sum of the first \( n \) terms of an A.P. be \( 3n^2 + 5n \). Then the sum of squares of the first \( 10 \) terms of the A.P. is:

Let \( a_1,a_2,a_3,a_4 \) be an A.P. of four terms such that each term of the A.P. and its common difference \( l \) are integers. If \( a_1+a_2+a_3+a_4=48 \) and \( a_1 a_2 a_3 a_4 + l^4 = 361 \), then the largest term of A.P. is equal to:

Consider an A.P.: \( a_{1}, a_{2}, \ldots , a_{n} ;\ a_{1}, > 0 \). If \( a_{2} - a_{1} = -\dfrac{3}{4} \), \( a_{n} = \dfrac{1}{4} a_{1} \), and \( \displaystyle \sum_{i=1}^{n} a_{i} = \dfrac{525}{2} \), then \( \displaystyle \sum_{i=1}^{17} a_{i} \) is equal to:

The sum \( 1+\dfrac{1}{2}(1^2+2^2)+\dfrac{1}{3}(1^2+2^2+3^2)+\ldots \) upto \( 10 \) terms is equal to:

The first term of an A.P. of \( 30 \) non-negative terms is \( \frac{10}{3} \). If the sum of this A.P. is the cube of its last term, then its common difference is:

In a G.P., if the product of the first three terms is \( 27 \) and the set of all possible values for the sum of its first three terms is \( \mathbb{R} - (a, b) \), then \( a^2 + b^2 \) is equal to __________.

The common difference of the A.P.: \( a_1, a_2, \ldots, a_m \) is \( 13 \) more than the common difference of the A.P.: \( b_1, b_2, \ldots, b_n \). If \( b_{31} = -277 \), \( b_{43} = -385 \) and \( a_{78} = 327 \), then \( a_1 \) is equal to:

Let \( a_1, a_2, a_3, \ldots \) be an A.P. and \( g_1 = a_1, g_2, g_3, \ldots \) be an increasing G.P. If \( a_1 = a_2 + g_2 = 1 \) and \( a_3 + g_3 = 4 \), then \( a_{10} + g_5 \) is equal to:

Let and Then the value of is equal to:

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JEE Main 2026 21st Jan Morning Shift

Let \( a_1 = 1 \) and for \( n \ge 1 \), \( a_{n+1} = \frac{1}{2} a_n + \frac{n^2 - 2n - 1}{n^2 (n+1)^2} \). Then \( \left| \sum_{n=1}^{\infty} \left( a_n - \frac{2}{n^2} \right) \right| \) is equal to _________.