Consider an infinite geometric series with first term $ a $ and common ratio $ r $. If the sum is $ 4 $ and the second term is $ \frac{3}{4} $,then find the values of $ a $ and $ r $.

The value of $\sum\limits_{r = 16}^{30} {(r + 2)(r - 3)}$ is equal to

The sum of the first $n$ terms of the series $\frac{1^{2}}{1}+\frac{1^{2}+2^{2}}{1+2}+\frac{1^{2}+2^{2}+3^{2}}{1+2+3}+\ldots$ is

If $\operatorname{gcd}(m, n) = 1$ and $1^2 - 2^2 + 3^2 - 4^2 + \ldots + (2021)^2 - (2022)^2 + (2023)^2 = 1012 m^2 n$,then $m^2 - n^2$ is equal to

If $1 + \sin \theta + \sin^2 \theta + \dots \text{ to } \infty = 4 + 2\sqrt{3}$,where $0 < \theta < \pi$ and $\theta \neq \frac{\pi}{2}$,then $\theta = $

Let $ABC$ be an equilateral triangle with side length $a$. $A$ new triangle is formed by joining the midpoints of all sides of the triangle $ABC$,and the same process is repeated infinitely many times. If $P$ is the sum of perimeters and $Q$ is the sum of areas of all the triangles formed in this process,then: