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

The sum of $n$ terms of an $\mathrm{A}.\mathrm{P}.$ is $2n+3n^2$, find the $r^{\mathrm{th}}$ term

If the sum of $m$ terms of an $\mathrm{A}.\mathrm{P}.$ is to the sum of $n$ terms as $m^2$ to $n^2$, show that the $m^{\mathrm{th}}$ term is to the $n^{\mathrm{th}}$ term as $2m-1$ is to $2n-1$ 

Prove that the sum of an odd number of terms in $\mathrm{A}.\mathrm{P}.$ is equal to the middle term multiplied by the number of terms 

If $S_n=n(5n-3)$ for all values of $n$, find the $p^{\mathrm{th}}$ term.

The number of terms in an $\mathrm{A}.\mathrm{P}.$ is even, the sum of the odd terms is $24$, of the even terms $30$, and the last term exceeds the first term by $10\frac{1}{2}$, find the number of terms.

There are two sets of numbers each consisting of three terms in $\mathrm{A}.\mathrm{P}.$ and the sum of each set is $15$. The common difference of the first set is greater by one than the common difference of the second set, and the product of the first set is to the product of the second set as $7$ to $8$, find the numbers.

Find the relation between $x$ and $y$ in order that the $r^{\mathrm{th}}$ mean between $x$ and $2y$ may be the same as the $r^{\mathrm{th}}$ mean between $2x$ and $y, n$ means being inserted in each case.

If the sum of an $\mathrm{A}.\mathrm{P}.$ is the same for $p$ as for $q$ terms, show that its sum for $p+q$ terms is zero.