An arithmetic progression has first term $a$ and common difference $d$. The sum of the first $100$ terms is $25$ times the sum of the first $20$ terms. Find $d$ in terms of $a$.

An arithmetic progression has first term $a$ and common difference $d$. The sum of the first $100$ terms is $25$ times the sum of the first $20$ terms. Write down an expression, in terms of $a$ , for the ${50}^{\mathrm{th}}$ term.

The tenth term of an arithmetic progression is $17$ and the sum of the first five terms is $190$. Find the first term of the progression and the common difference.

The tenth term of an arithmetic progression is $17$ and the sum of the first five terms is $190$. Given that the $n^{\mathrm{th}}$ term of the progression is $-19$, find the value of $n$.

The fifth term of an arithmetic progression is $18$ and the sum of the first eight terms is $186$ . Find the first term and the common difference.

The first term of a geometric progression is $32$ and the fourth term is $\frac{1}{2}$. Find the sum to infinity of the progression.

The seventh term of an arithmetic progression is $19$ and the sum of the first twelve terms is $224$ . Find the fourth term.

A geometric progression has first term $3$ and common ratio $r$. A second geometric progression has first term $2$ and common ratio $\frac{1}{5}r$. The two progressions have the same sum to infinity $S$. Find the value of $r$ and the value of $S$.

A geometric progression has first term $a$, common ratio $r$ and sum to infinity $S$. A second geometric progression has first term $5a$ , common ratio $3r$ and sum to infinity $10S$. Find the value of $r$.[Write your answer as decimal]