List of top Mathematics Questions on Arithmetic and Geometric Progressions

Let \( T_r \) be the \( r^{th} \) term of an A.P. If for some \( m \), \( T_m = \frac{1}{25} \), \( T_{25} = \frac{1}{20} \), and \( \sum_{r=1}^{25} T_r = 13 \), then

\[ 5m \sum_{r=m}^{2m} T_r \text{ is equal to:} \]

If \( 7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5 + 3\alpha) + \cdots \), then the value of \( \alpha \) is:

Let \( T_r \) be the \( r^{\text{th}} \) term of an A.P. If for some \( m \), \( T_m = \dfrac{1}{25} \), \( T_{25} = \dfrac{1}{20} \), and \( \displaystyle\sum_{r=1}^{25} T_r = 13 \), then \( 5m \displaystyle\sum_{r=m}^{2m} T_r \) is equal to:

If \( 7 = 5 + \frac{1}{7}(5 + \alpha) + \frac{1}{7^2}(5 + 2\alpha) + \frac{1}{7^3}(5 + 3\alpha) + \cdots \), then the value of \( \alpha \) is:
Let \( S_n = \frac{1}{2} + \frac{1}{6} + \frac{1}{12} + \frac{1}{20} + \dots \) up to \( n \) terms. If the sum of the first six terms of an A.P. with first term \( -p \) and common difference \( p \) is \( \sqrt{2026 S_{2025}} \), then the absolute difference between the 20th and 15th terms of the A.P. is:
The value of \[ \lim_{n \to \infty} \left( \sum_{k=1}^{n} \frac{k^3 + 6k^2 + 11k + 5}{(k + 3)!} \right) \] is:
If a, b, c are in increasing A.P. and a + 1, b, c + 3 are in G.P. If A.M. of a, b, c is 8. Find cube of G.M. of a, b, c.