The largest term in the sequence $a_n = \frac{n^2}{n^3 + 200}$ is given by

The sum of all $3$-digit numbers less than or equal to $500$,formed without using the digit $1$,which are also multiples of $11$,is ..... .

If $a, b, c$ are in $G.P.$,and $a - b, c - a, b - c$ are in $H.P.$,then $a + 4b + c$ is equal to

Let $a_1, a_2, a_3, \ldots$ be a sequence of positive integers in arithmetic progression with common difference $2$. Also,let $b_1, b_2, b_3, \ldots$ be a sequence of positive integers in geometric progression with common ratio $2$. If $a_1 = b_1 = c$,then the number of all possible values of $c$,for which the equality $2(a_1 + a_2 + \ldots + a_n) = b_1 + b_2 + \ldots + b_n$ holds for some positive integer $n$,is:

Let $A_1$ and $A_2$ be two arithmetic means and $G_1, G_2, G_3$ be three geometric means between two distinct positive numbers $a$ and $b$. Then $G_1^4 + G_2^4 + G_3^4 + G_1^2 G_3^2$ is equal to

Write the first five terms of the sequence defined by $a_{1} = -1$ and $a_{n} = \frac{a_{n-1}}{n}$ for $n \geq 2$,and obtain the corresponding series.