Quick questions on Sequences and series: arithmetic, geometric, sigma notation and convergence - OCR A-Level Maths A

An arithmetic sequence has a constant common difference $d$. The $n$th term and the sum are:

A geometric sequence has a constant common ratio $r$. The $n$th term and the sum are:

A common Paper 1 task combines sigma notation with the standard sum formulae. The key facts are $\sum_{r=1}^{n} 1 = n$ and $\sum_{r=1}^{n} c = cn$ for a constant $c$. Combined with the arithmetic sum, these let you evaluate sums written in sigma form without listing every term.

Sequences model real situations such as repayments, salaries that rise by a fixed percentage, or the bounce heights of a ball. A fixed yearly increase gives an arithmetic model; a fixed percentage change gives a geometric model. Identifying which model applies, then choosing the $n$th-term or sum formula, is a frequent overarching-theme OT3 application. For a geometric decay such as a ball losing a fixed fraction of its height on each bounce, the total distance travelled is found with the sum to infinity when the ratio satisfies $|r| < 1$.

Arithmetic uses $(n - 1)d$ and geometric uses $r^{n-1}$, not $nd$ or $r^n$.

Find the 20th term of the arithmetic sequence $5, 8, 11, \dots$. [2 marks]

A geometric series has $a = 12$, $r = \tfrac{1}{3}$. Find its sum to infinity. [2 marks]