H S Hall and S R Knight Solutions for Chapter: Harmonical Progression, Theorems Connected with The Progression, Exercise 1: EXAMPLES. VI.a.

Find the sum of the $n$ terms of the series whose $n^{\mathrm{th}}$ term is $3\left(4^n+2n^2\right)-4n^3$.

If the $(m+1{)}^{\mathrm{th}},(n+1{)}^{\mathrm{th}}$ and $(r+1{)}^{\mathrm{th}}$ terms of an $\mathrm{A}.\mathrm{P}.$ are in $\mathrm{G}.\mathrm{P}.$ and $m,n$ and $r$ are in $\mathrm{H}.\mathrm{P}.$, show that the ratio of the common difference to the first term of the $\mathrm{A}.\mathrm{P}.$ is $-\frac{2}{n}$.

If $l,m,n$ are three numbers in $\mathrm{G}.\mathrm{P}.,$ prove that the ratio of the first term of an $\mathrm{A}.\mathrm{P}.$ whose $l^{th}, m^{th}$ and $n^{th}$ terms are in $\mathrm{H}.\mathrm{P}.$ to the common difference is $\left(m+1\right):1$.

If the sum of $n$ terms of a series is $a+bn+c{n}^2$. Find the $n^{th}$ term and the nature of the series.

Find the sum of $n$ terms of the series whose $n^{th}$ term is $4n\left(n^2+1\right)-\left(6n^2+1\right)$.

Between any two quantities, let there be inserted two arithmetic means $A_1 \& A_2,$ two geometric means $G_1 \& G_2$ and two harmonic means $H_1$ $\& H_2$. Show that $G_1{G}_2:H_1{H}_2=\left(A_1+A_2\right):\left(H_1+H_2\right)$.

If $p$ be the first of $n$ arithmetic means between two numbers and $q$ be the first of $n$ harmonic means between the same two numbers, prove that the value of $q$ cannot lie between $p$ and ${\left(\frac{n+1}{n-1}\right)}^{2}p.$

Find the sum of the cubes of the terms of an $\mathrm{A}.\mathrm{P}.$ and show that it is exactly divisible by the sum of the terms.