The first and last terms of an $AP$ are $75$ and $375$ respectively. If its sum is $3600$,find the number of terms in the $AP$ and its common difference.

$\sqrt {\underbrace {111........1}_{200\,digits} - \underbrace {222.......2}_{100\,digits}} $ equals :-

There are two such pairs of non-zero real values of $a$ and $b$,i.e.,$(a_1, b_1)$ and $(a_2, b_2)$,for which $2a+b, a-b, a+3b$ are three consecutive terms of a $G.P.$. Then the value of $2(a_1b_2 + a_2b_1) + 9a_1a_2$ is-

The product of three consecutive terms of a $G.P.$ is $512$. If $4$ is added to each of the first and the second of these terms,the three terms now form an $A.P.$ Then the sum of the original three terms of the given $G.P.$ is

The sum of $n$ terms of the series whose $n^{th}$ term is $n(n + 1)$ is equal to

Let $a_n$ be a sequence such that $a_1 = 5$ and $a_{n+1} = a_n + (n - 2)$ for all $n \in N$. Then $a_{51}$ is: