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 $b$.

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.