A certain number of people contributed to a charity. The first person contributed one rupee. The rule for contribution was that the next person would contribute double the amount already raised. If the total money raised for the charity was 2187 rupees, then the number of people who contributed to the charity is

The numbers $-16,2^{x+3}-2^{2 x-1}-16,2^{2 x-1}+16$ are in an arithmetic progression. Then $x$ equals ________.

A new sequence is obtained from the sequence of positive integers $(1,2,3, \ldots)$ by deleting all the perfect squares. Then the $2022^{\text {nd }}$ term of the new sequence is ________.

Let $S_1 = \{100, 105, 110, 115, ... \}$ and $S_2 = \{100, 95, 90, 85, ... \}$ be two series in arithmetic progression. If $a_k$ and $b_k$ are the $k$-th terms of $S_1$ and $S_2$, respectively, then $\sum_{k=1}^{20} a_k b_k$ equals __________.

The $3^{\text {rd }}, 14^{\text {th }}$ and $69^{\text {th }}$ terms of an arithmetic progression form three distinct and consecutive terms of a geometric progression. If the next term of the geometric progression is the $n^{\text {th }}$ term of the arithmetic progression, then $n$ equals ________.

It is given that the sequence {$x_n$} satisfies $x_1 = 0, x_{n+1} = x_n + 1 + 2√(1+x_n)$ for $n = 1,2,...$ Then $x_{31}$ is _______

The sum of the first 15 terms in an arithmetic progression is 200, while the sum of the next 15 terms is 350. Then the common difference is