Question 1

What is the difference between a sequence and a series?

Accepted Answer

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.

Question 2

When do I use $S_n = \frac{a(1-r^n)}{1-r}$ versus $S_n = \frac{a(r^n-1)}{r-1}$?

Accepted Answer

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 $|r|>1$.

Question 3

Why is the binomial expansion only valid for $|x|<1$ when n is not a positive integer?

Accepted Answer

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).

Question 4

How can I tell if a problem involves an AP or a GP?

Accepted Answer

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 $y^2 = xz$).

Series

What you need to know

Building with Patterns