Series — common mistakes

In exam questions, you'll often need to solve simultaneous equations involving the formulae for terms and sums. Be methodical. Write down the formulae you are using and substitute the given information carefully. Also, always check the validity condition for sums to infinity ($|r|<1$) and binomial expansions.

A sequence is an ordered list of numbers, like $u_1, u_2, u_3, \dots$. A series is the sum of the terms in a sequence, like $u_1 + u_2 + u_3 + \dots$. Cambridge questions often use 'progression' and 'sequence' interchangeably.

Both formulae are correct and will give the same answer. However, to make calculations easier and avoid negative signs, it's conventional to use $S_n = \frac{a(1-r^n)}{1-r}$ when $|r|<1$ and $S_n = \frac{a(r^n-1)}{r-1}$ when $|r|>1$.

When $n$ is not a positive integer, the expansion is an infinite series. For the sum of this series to converge to a finite value (equal to $(1+x)^n$), the terms must get smaller and smaller. This only happens if $|x|<1$. If $|x| \ge 1$, the terms get larger or stay the same size, and the sum diverges (goes to infinity).

Look for keywords. 'Common difference' implies an AP. 'Common ratio' implies a GP. If you're given three consecutive terms, say $x, y, z$: for an AP, $y-x = z-y$ (or $2y = x+z$); for a GP, $y/x = z/y$ (or $y^2 = xz$).