Consider the following sequence.
$$\frac{1}{1} \big\vert \; \frac{1}{2}, \frac{2}{1} \big\vert \; \frac{1}{3}, \frac{2}{2}, \frac{3}{1} \big\vert \; \frac{1}{4}, \frac{2}{3}, \frac{3}{2}, \frac{4}{1} \big\vert \; \frac{1}{5}, \frac{2}{4}, \frac{3}{3}, \frac{4}{2}, \frac{5}{1} \big\vert \; \cdots$$
The terms are divided into groups such that the sum of the denominator and numerator of every term in the group is equal and the numerators are in ascending order.
Let $m$ and $n$ be positive integers such that $m \ge n$.
a) Express the $n^{th}$ term in the $m^{th}$ group using $m$ and $n$.
b) Express the term number of the $n^{th}$ term in the $m^{th}$ group using $m$ and $n$. For example, the $4^{th}$ term in the $5^{th}$ group is $\frac{4}{2}$ and the term number is $14$. Write your answer as a single fraction.

For an arithmetic sequence, the $m$th term is $p$ and the $n$th term is $q$. Find the $\left(m+n\right)$th term in terms of $m, n, p$ and $q$. Give your answer as a single fraction.

The numbers $1,a,b$ form an arithmetic sequence and $1,a,b^2$ forms a geometric sequence. If $a \ne b$, find $a$ and $b$.

What number comes next in the following sequence of positive integers?
$$1,2,4,8,61,23,46,821,652,\ldots$$

Consider two sequences:

$4, 7, 10, 13, 16,\dots$

$1000, 995, 990, 985, 980,\dots$

Find the sum of all the terms common to both sequences.

For a given arithmetic sequence with positive common difference, $\displaystyle \sum_{k=3}^{7}u_k=20$ and $\displaystyle \sum_{k=4}^{7}\left(u_k\right)^2=120$.
Find the general term.

The sum of an infinite geometric series is a positive number $S_\infty$ and the second term in the series is $1$. What is the smallest possible value of $S_\infty?$

The sum of the first $17$ terms of an arithmetic sequence is $714$.
Find the 9th term of the sequence.