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. 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?
$$1,2,4,8,61,23,46,821,652,\ldots$$