TMUA Sequences and Series

Arithmetic series

With first term $a$ and common difference $d$: $u_n=a+(n-1)d$ and $S_n=\tfrac n2\left(2a+(n-1)d\right)$.

Geometric series

With first term $a$ and common ratio $r$: $u_n=ar^{n-1}$ and $S_n=\dfrac{a(1-r^n)}{1-r}$. If $|r|<1$ the sum to infinity is $\dfrac{a}{1-r}$.

Recurrence relations

For $u_{n+1}=f(u_n)$, a limit $L$ (if it exists) must satisfy $L=f(L)$.

Spotting the type

Check differences first. A constant difference means arithmetic, a constant ratio means geometric, and constant second differences mean a quadratic sequence.

The sum of the first $n$ natural numbers is $\dfrac{n(n+1)}2$. A geometric series with $|r|\ge1$ does not converge, so never use $\dfrac a{1-r}$ there. Read the first few terms before choosing a formula.

For an arithmetic series, writing the sum forwards and backwards gives the pairing idea behind the formula: every pair of terms from the two ends has the same total.

Check small cases: find the first two or three terms from your formula and compare them with the information in the question before you move on.

Worked examples

Common mistakes

• Using $n$ instead of $n-1$ in $a+(n-1)d$ or $ar^{n-1}$.
• Using the sum to infinity formula when $|r|\ge1$.
• Mixing up the number of terms with the position of the last term.
• Assuming a recurrence has a limit without checking it converges.

Test yourself on this topic with our free practice questions.