The interior angles of a polygon with \( n \) sides, are in A.P. with common difference \( 6^\circ \). If the largest interior angle of the polygon is \( 219^\circ \), then \( n \) is equal to __________________.
Let \( a_1,a_2,\ldots,a_{2024} \) be an Arithmetic Progression such that \( a_1+(a_5+a_{10}+a_{15}+\cdots+a_{2020})+a_{2024}=2233 \). Then \( a_1+a_2+a_3+\cdots+a_{2024} \) is equal to ________.
In the expansion of \( \left( \sqrt[3]{2} + \frac{1}{\sqrt[3]{3}} \right)^{n} \), \( n \in \mathbb{N} \), if the ratio of \( 15^{\text{th}} \) term from the beginning to the \( 15^{\text{th}} \) term from the end is \( \frac{1}{6} \), then the value of \( {}^{n}C_{3} \) is:
The sum \( 1+3+11+25+45+71+\cdots \) upto \( 20 \) terms, is equal to:
Let \( a_1, a_2, a_3, \ldots \) be a G.P. of increasing positive numbers. If \( a_3 a_5 = 729 \) and \( a_2 + a_4 = \frac{111}{4} \), then \( 24(a_1 + a_2 + a_3) \) is equal to:
Let \( x_1, x_2, x_3, x_4 \) be in a geometric progression, If \( 2, 7, 9, 5 \) are subtracted respectively from \( x_1, x_2, x_3, x_4 \), then the resulting numbers are in an arithmetic progression. Then the value of \( \dfrac{1}{24} (x_1 x_2 x_3 x_4) \) is:
Let \(\displaystyle f(x)=\lim_{n\to\infty}\sum_{r=0}^{n}\left(\dfrac{\tan\left(\frac{x}{2^{r+1}}\right)+\tan^{3}\left(\frac{x}{2^{r+1}}\right)}{1-\tan^{2}\left(\frac{x}{2^{r+1}}\right)}\right) \). Then \(\displaystyle \lim_{x\to 0}\dfrac{e^{x}-e^{f(x)}}{(x-f(x))} \) is equal to __________________.
In an arithmetic progression, if \( S_{40} = 1030 \) and \( S_{12} = 57 \), then \( S_{30} - S_{10} \) is equal to:
Let \( A(4,-2) \), \( B(1,1) \) and \( C(9,-3) \) be the vertices of a triangle \( ABC \). Then the maximum area of the parallelogram \( AFDE \), formed with vertices \( D \), \( E \) and \( F \) on the sides \( BC \), \( CA \) and \( AB \) of the triangle \( ABC \) respectively, is __________________.
Let \( a_{n} \) be the \( n^{\text{th}} \) term of an A.P. If \( S_{n}=a_{1}+a_{2}+a_{3}+\cdots +a_{n}=700 \), \( a_{6}=7 \) and \( S_{7}=7 \) then \( a_{n} \) is equal to:
If \( \dfrac{1}{1^4} + \dfrac{1}{2^4} + \dfrac{1}{3^4} + \ldots \infty = \dfrac{\pi^4}{90} \), \( \dfrac{1}{1^4} + \dfrac{1}{3^4} + \dfrac{1}{5^4} + \ldots \infty = \alpha \), \( \dfrac{1}{2^4} + \dfrac{1}{4^4} + \dfrac{1}{6^4} + \ldots \infty = \beta \), then \( \dfrac{\alpha}{\beta} \) is equal to:
Let \( T_r \) be the \( r^{th} \) term A.P. If for some, \( m \), \( T_m = \frac{1}{25} \), \( T_{25} = \frac{1}{20} \) and \( 20 \displaystyle \sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle \sum_{r=m}^{2m} T_r \) is equal to:
Let \( a_{1},a_{2},a_{3},\ldots \) be in an A.P. such that \( \displaystyle \sum_{k=1}^{12} 2a_{2k-1} = -\frac{72}{5} a_{1},\ a_{1} \ne 0 \). If \( \displaystyle \sum_{k=1}^{n} a_{k} = 0 \), then \( n \) is:
Let \( a_1, a_2, a_3, \ldots \) be a G.P. of increasing positive terms. If \( a_1 a_5 = 28 \) and \( a_2 + a_4 = 29 \), then \( a_6 \) is equal to:
Consider an A.P. of positive integers, whose sum of the first three terms is \( 54 \) and the sum of the first twenty terms lies between \( 1600 \) and \( 1800 \). Then its \( 11^{\text{th}} \) term is:
If $ \displaystyle \sum_{r=1}^{n} T_r = \frac{(2n - 1)(2n + 1)(2n + 3)(2n + 5)}{64}$$, \: then\:\: $$ \displaystyle \lim_{n \to \infty} \displaystyle \sum_{r=1}^{n} \left( \frac{1}{T_r} \right)$ is equal to:
The sum \( 1 + \frac{1+3}{2!} + \frac{1+3+5}{3!} + \frac{1+3+5+7}{4!} + \ldots \) upto \( \infty \) terms, is equal to:
Consider two sets \( A \) and \( B \), each containing three numbers in A.P. Let the sum and the product of the elements of \( A \) be \( 36 \) and \( p \) respectively and the sum and the product of the elements of \( B \) be \( 36 \) and \( q \) respectively. Let \( d \) and \( D \) be the common differences of AP's in \( A \) and \( B \) respectively such that \( D = d + 3 \) and \( d > 0 \). If \( \dfrac{p+q}{p-q} = \dfrac{19}{5} \), then \( p - q \) is equal to
Let ⟨ \( a_n \) ⟩ be a sequence such that \( a_0 = 0 \), \( a_1 = \frac{1}{2} \) and \( 2a_{n+2} = 5a_{n+1} - 3a_n \), \( n = 0, 1, 2, 3, \ldots \).Then \( \displaystyle \sum_{k=1}^{100} a_k \) is equal to:
Suppose that the number of terms in an A.P. is \( 2k , \; k \in \mathbb{N} \). If the sum of all odd terms of the A.P. is \( 40 \), the sum of all even terms is \( 55 \) and the last term of the A.P. exceeds the first term by \( 27 \), then \( k \) is equal to:
If \( 7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5+ 3\alpha) + \ldots \\infty \), then the value of \( \alpha \) is:
If the sum of the first \( 10 \) terms of the series \( \dfrac{4 \cdot 1}{1 + 4 \cdot 1^{4}} + \dfrac{4 \cdot 2}{1 + 4 \cdot 2^{4}} + \dfrac{4 \cdot 3}{1 + 4 \cdot 3^{4}} + \ldots \) is \( \dfrac{m}{n} \), where \( \gcd(m, n) = 1 \), then \( m + n \) is equal to __________.
For positive integers \( n \), if \( 4a_n=(n^2+5n+6 )\) and \(\displaystyle S_n=\sum_{k=1}^{n}\left(\frac{1}{a_k}\right) \), then the value of \( 507\,S_{2025} \) is:
