If $x$ is added to each of numbers $3, 9, 21$ so that the resulting numbers may be in $G.P.$, then the value of $x$ will be
$3$
$\frac{1}{2}$
$2$
$\frac{1}{3}$
If $a,\;b,\;c,\;d$ and $p$ are different real numbers such that $({a^2} + {b^2} + {c^2}){p^2} - 2(ab + bc + cd)p + ({b^2} + {c^2} + {d^2}) \le 0$, then $a,\;b,\;c,\;d$ are in
Consider an infinite $G.P. $ with first term a and common ratio $r$, its sum is $4$ and the second term is $3/4$, then
If the sum of the series $1 + \frac{2}{x} + \frac{4}{{{x^2}}} + \frac{8}{{{x^3}}} + ....\infty $ is a finite number, then
The sum of a $G.P.$ with common ratio $3$ is $364$, and last term is $243$, then the number of terms is
Find the sum of first $n$ terms and the sum of first $5$ terms of the geometric
series $1+\frac{2}{3}+\frac{4}{9}+\ldots$