The Fibonacci sequence is defined by $a_1 = 1, a_2 = 1$ and $a_n = a_{n-1} + a_{n-2}$ for $n > 2$. Find $\frac{a_{n+1}}{a_n}$ for $n = 1, 2, 3, 4, 5$.

The sum of the first $9$ terms of the series $\frac{1^3}{1} + \frac{1^3 + 2^3}{1 + 3} + \frac{1^3 + 2^3 + 3^3}{1 + 3 + 5} + \dots$ is:

Find the sum of the following series up to $n$ terms:
$\frac{1^{3}}{1}+\frac{1^{3}+2^{3}}{1+3}+\frac{1^{3}+2^{3}+3^{3}}{1+3+5}+\ldots$

The value of the sum $1 \times 2 \times 3 + 2 \times 3 \times 4 + 3 \times 4 \times 5 + \ldots$ up to $n$ terms is equal to

Let $75 \ldots 57$ denote the $(r+2)$ digit number where the first and the last digits are $7$ and the remaining $r$ digits are $5$. Consider the sum $S = 77 + 757 + 7557 + \ldots + 75 \ldots 57$ (where the last term has $98$ digits). If $S = \frac{75 \ldots 57 + m}{n}$,where $m$ and $n$ are natural numbers less than $3000$,then the value of $m + n$ is:

In the sequence of sets $(1,2,3), (4,5,6), (7,8,9,10), \ldots$,the sum of elements in the $50^{th}$ set is