Recurrence relation worksheet

Recurrence relation at a glance

A recurrence relation is a relation which creates a sequence of numbers. Consecutive terms of a sequence are found using a rule which gives the next term based on the current term. An example of a linear recurrence relation is $u_{n+1}=2u_{n}+3$. If the first term is 5, the next terms of the sequence  are found by multiplying each previous term by 2 and then adding 3.

$u_{0}=5$

$u_{1}=2×5+3=13$

$u_{2}=2×13+3=29$ etc. . .

We might be asked to find an expression for $u_{n}$. This is the nth term formula for the sequence. A recurrence relation of the form $u_{n+1}=u_{n}+a$ where the coefficient of $u_{n}$ is 1 and a is a constant, will produce a linear sequence. An example of this is the following recurrence relation: $u_{n+1}=u_{n}+5$ with initial value $u_{0}=1$, which gives the following sequence: 1, 6, 11, 16, 21, …. In this case, we can find $u_{n}=5n-4$.

Looking forward, students can then progress to additional sequences worksheets or more  algebra worksheets, for example a sequences worksheet, or simultaneous equations worksheet.

For more teaching and learning support on Algebra our GCSE maths lessons provide step by step support for all GCSE maths concepts.