For a sequence $(t_n)$,if $S_n = 5(2^n - 1)$,then $t_n = \ldots$

If $a, b$ and $c$ form a geometric progression with common ratio $r$,then the sum of the ordinates of the points of intersection of the line $ax + by + c = 0$ and the curve $x + 2y^2 = 0$ is

Find the Geometric Mean $(G.M.)$ of the sequence $1, 2, 2^2, \dots, 2^n$.

The three sides of a right-angled triangle are in $GP$ (geometric progression). If the two acute angles are $\alpha$ and $\beta$,then $\tan \alpha$ and $\tan \beta$ are

Consider two $G$.Ps. $2, 2^{2}, 2^{3}, \ldots$ and $4, 4^{2}, 4^{3}, \ldots$ of $60$ and $n$ terms respectively. If the geometric mean of all the $60+n$ terms is $(2)^{\frac{225}{8}}$,then $\sum_{k=1}^{n} k(n-k)$ is equal to.

If the $10^{th}$ term of a geometric progression is $9$ and the $4^{th}$ term is $4$,then its $7^{th}$ term is: