If the first three terms of the sequence $\frac{1}{16}, a, b, \frac{1}{6}$ are in a geometric progression and the last three terms are in a harmonic progression,then the values of $a$ and $b$ are:

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 ..... .

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:

Write the first three terms in each of the following sequences defined by the following: $a_{n} = \frac{n-3}{4}$

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

Let $a_n = \frac{10^n}{n!}$ for $n = 1, 2, 3, \ldots$. The greatest value of $n$ for which $a_n$ is the greatest is: