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 ________.

There are numbers $a_1, a_2, a_3, \ldots, a_n$ each of them being $+1$ or $-1$. If it is known that $a_1 a_2 + a_2 a_3 + a_3 a_4 + \ldots a_{n-1} a_n + a_n a_1 = 0$ then

Given that $1 + \frac{1}{2^2} + \frac{1}{3^2} + \frac{1}{4^2} + ... = \frac{\pi^2}{6}$, the value of $1 + \frac{1}{3^2} + \frac{1}{5^2} + \frac{1}{7^2} + ...$ 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

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

Gita starts from point A and walks 1000 m east. She then walks 800 m north, followed by 640 m west and 512 m south, reaching point B. After this, she continues moving in the same cyclic order: east, north, west, south, with each successive movement 20% shorter than the previous one. After infinitely many such moves, approximately how far in meters will Gita be from her starting point A?