The $9^{th}$ term of the sequence $27, 9, 5\frac{2}{5}, 3\frac{6}{7}, \dots$ is $.....$

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:

Find the sum of the series $1 + (1 + 2) + (1 + 2 + 3) + \dots$ up to $n$ terms.

If $0 < \theta, \phi < \frac{\pi}{2}$,$x = \sum_{n=0}^{\infty} \cos^{2n} \theta$,$y = \sum_{n=0}^{\infty} \sin^{2n} \phi$,and $z = \sum_{n=0}^{\infty} \cos^{2n} \theta \cdot \sin^{2n} \phi$,then:

Find the sum of $n$ terms of the series $1 + 6 + 13 + 22 + 33 + \dots$

If $\sum\limits_{r=1}^\infty \frac{1}{(2r-1)^2} = \frac{\pi^2}{8}$,then $\sum\limits_{r=1}^\infty \frac{1}{r^2} = \dots$