H S Hall and S R Knight Solutions for Chapter: Geometrical Progression, Exercise 2: Examples. V.b.

Find the sum of the series $\frac{4}{7}-\frac{5}{7^2}+\frac{4}{7^3}-\frac{5}{7^4}+\frac{4}{7^5}-\frac{5}{7^6}+\dots$ to infinity.

If $a, b, c, d$ be in $\mathrm{G}.\mathrm{P}.$, prove that $(b-c{)}^2+(c-a{)}^2+(d-b{)}^2=(a-d{)}^2$.

If the arithmetic mean between $a$ and $b$ is twice as great as the geometric mean, show that $a:b=2+\sqrt{3}:2-\sqrt{3}$

Find the sum of $n$ terms of the series the $r\mathrm{th}$ term of which is $(2r+1)2^r$.

Find the sum of $2n$ terms of a series of which every even term is $a$ times the term before it, and every odd term is $c$ times the term before it, the first term being unity.

If $S_n$ denotes the sum of $n$ terms of a $\mathrm{G}.\mathrm{P}.$ whose first term is $a$ and common ratio $r$, find the sum of $S_1, S_3, S_5,\dots \dots \dots S_{2n-1}$.

If $S_1, S_2, S_2, S_3,\dots S_p$ are the sum of infinite geometric series whose first terms are $1, 2, 3,\dots ,p$ and whose common ratios are $\frac{1}{2}, \frac{1}{3}, \frac{1}{4},\dots ,p+1$ respectively. Then, prove that $S_1+S_2+S_3+\dots +S_p=\frac{p}{2}(p+3)$

If $r<1$ and positive, and $m$ is a positive integer, show that $(2m+1)r^m(1-r)<1-r^{2m+1}$. Hence, show that $n{r}^n$ is indefinitely small when $n$ is indefinitely great.