Quick questions on Arithmetic progressions explained: H2 Mathematics Sequences and Series

If you are given $S_n$ as a formula in $n$, the $n$th term is $u_n = S_n - S_{n-1}$. For an AP this expression is always linear in $n$, and the coefficient of $n$ is the common difference.

Many H2 problems describe an AP without naming it, so the first skill is spotting the constant common difference. Any situation where a quantity increases or decreases by the same fixed amount each step, equal monthly repayments, seats increasing by a fixed number per row, a salary rising by a set raise each year, is arithmetic. Once you identify $a$ (the starting value) and $d$ (the fixed change), the whole problem reduces to substituting into the two AP formulae. Translating the words into $a$ and $d$ before reaching for a formula is what turns a wordy question into a routine calculation.

The terms from the $p$th to the $q$th number $q - p + 1$, not $q - p$.

Find the $12$th term of the AP $7, 10, 13, \ldots$ [2 marks]

The sum of the first $n$ terms of an AP is $S_n = 3n^2 - n$. Find the first term and the common difference. [3 marks]

An AP has $a = 100$ and $d = -4$. Find how many terms are positive. [3 marks]