Sequence & Series (AP, GP, HP)

Consider the sequence $$ t_1 = 1 $$, $$ t_2 = -1 $$, and $$ t_n = \left(\frac{n-3}{n-1}\right) t_{n-2} \quad \text{fo...

Suppose $$ x_1, x_2, x_3, \dots, x_{100} $$ are in arithmetic progression such that $$ x_5 = -4 $$ and $$ 2x_6 + 2x_9 = ...

For any natural number $$ n $$, suppose the sum of the first $$ n $$ terms of an arithmetic progression is $$ S_n = n...

The sum of the infinite series $$ \frac{1}{5} \left( \frac{1}{5} - \frac{1}{7} \right) + \left( \frac{1}{5} \right)^2...

The value of $$ 1 + \left(1 + \frac{1}{3}\right)\frac{1}{4} + \left(1 + \frac{1}{3} + \frac{1}{9}\right)\frac{1}{16} ...

The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is ...

On day one, there are 100 particles in a laboratory experiment. On day $$ n $$, where $$ n \geq 2 $$, one out of every ...

Let both the series $$a_1, a_2, a_3, \dots$$ and $$b_1, b_2, b_3, \dots$$ be in arithmetic progression such that the co...

The average of a non-decreasing sequence of $$ N $$ numbers $$ a_1, a_2, \dots, a_N $$ is 300. If $$ a_1 $$ is replaced...

Let $$ a_n = 46 + 8n $$ and $$ b_m = 98 + 4m $$ be two sequences for natural numbers $$ n \leq 100 $$. Then, the sum of...

Let $$a_n$$ and $$b_n$$ be two sequences such that $$a_n = 13 + 6(n-1)$$ and $$b_n = 15 + 7(n-1)$$ for all natural numb...

A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the nu...

For some positive and distinct real numbers $$ x $$, $$ y $$, and $$ z $$, if $$ \frac{1}{\sqrt{x} + \sqrt{z}} $$ is ...