If $1+\cos x+\cos ^2 x+\cos ^3 x+\ldots \text{ to } \infty = 4+2 \sqrt{3}$,then $x=$

If $a, b, c$ and $d$ are in $G.P.$,show that:
$(a^{2}+b^{2}+c^{2})(b^{2}+c^{2}+d^{2})=(ab+bc+cd)^{2}$

If the sum of an infinite $GP$ $a, ar, ar^{2}, ar^{3}, \ldots$ is $15$ and the sum of the squares of its each term is $150$,then the sum of $ar^{2}, ar^{4}, ar^{6}, \ldots$ is:

$A$ particle starts at the origin and moves $1$ unit horizontally to the right and reaches $P_{1}$, then it moves $\frac{1}{2}$ unit vertically up and reaches $P_{2}$, then it moves $\frac{1}{4}$ unit horizontally to the right and reaches $P_{3}$, then it moves $\frac{1}{8}$ unit vertically down and reaches $P_{4}$, then it moves $\frac{1}{16}$ unit horizontally to the right and reaches $P_{5}$ and so on. Let $P_{n} = (x_{n}, y_{n})$ and $\lim_{n \rightarrow \infty} x_{n} = \alpha$ and $\lim_{n \rightarrow \infty} y_{n} = \beta$. Then, $(\alpha, \beta)$ is

$A$ person writes a letter to four of his friends. He asks each one of them to copy the letter and mail it to four different persons with the instruction that they continue the chain similarly. Assuming that the chain is not broken and that it costs $50$ paise to mail one letter,find the total amount spent on postage when the $8^{\text{th}}$ set of letters is mailed.

The two geometric means between the numbers $1$ and $64$ are