If $3 + 3\alpha + 3{\alpha ^2} + ………\infty = \frac{{45}}{8}$, then the value of $\alpha $ will be

The sum of two numbers is $6$ times their geometric mean, show that numbers are in the ratio $(3+2 \sqrt{2}):(3-2 \sqrt{2})$

The geometric series $a + ar + ar^2 + ar^3 +….. \infty$ has sum $7$ and the terms involving odd powers of $r$ has sum $'3'$, then the value of $(a^2 -r^2)$ is –

If ${G_1}$ and ${G_2}$ are two geometric means and $A$ the arithmetic mean inserted between two numbers, then the value of $\frac{{G_1^2}}{{{G_2}}} + \frac{{G_2^2}}{{{G_1}}}$is

The sum to infinity of the progression $9 – 3 + 1 – \frac{1}{3} + …..$ is