If $\alpha_r$ and $\beta_r$ (where $\alpha_r < \beta_r$) are the roots of the quadratic equation $x^2 - r^2(r + 1)x + r^5 = 0$,then find the value of $\sum_{r=1}^{n} (3\alpha_r + 2\beta_r)$.

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:

Let $a_{1}=b_{1}=1$,$a_{n}=a_{n-1}+2$,and $b_{n}=a_{n}+b_{n-1}$ for every natural number $n \geq 2$. Then $\sum_{n=1}^{15} a_{n} \cdot b_{n}$ is equal to $.........$

If the $n^{th}$ term of a sequence is $n(n + 1)$,then what is the sum of its $n$ terms?

Find the sum to $n$ terms of the sequence,$8, 88, 888, 8888, \ldots$

$1 + \frac{1^3 + 2^3}{1 + 2} + \frac{1^3 + 2^3 + 3^3}{1 + 2 + 3} + \dots + \frac{1^3 + 2^3 + 3^3 + \dots + 15^3}{1 + 2 + 3 + \dots + 15} - \frac{1}{2}(1 + 2 + 3 + \dots + 15)$ is equal to