Solving Recurrence Relations

Video lesson · from 7:461 worked example

One lesson video covers all of Recurrence Relations, so it opens at 7:46 for this topic — not from the beginning.

Theory

If a sequence is defined by a linear recurrence relation $U_{n+1} = aU_n + b$ with unknowns $a$ and $b$ , but we know several terms of the sequence, then we can find the values of $a$ and $b$ .

We can then solve for $a$ and $b$ using simultaneous equations.

⚠️ Common Examiner Traps

Two unknowns need two equations: use two consecutive pairs of terms to form simultaneous equations in $a$ and $b$ .
Subtract to eliminate $b$ : since $b$ appears with the same coefficient in both, subtracting one equation from the other leaves $a$ alone.
Substitute back for $b$ : and check both original equations, which catches arithmetic errors immediately.
Keep lines consistent: write each equation out in full rather than scoring terms out, so every line follows from the one above.

Worked examples

Example 1

A sequence is defined by $U_{n+1} = aU_n + b$ with

$U_1 = 4,\ U_2 = 3.6 \text{ and } U_3 = 2.04$ .

Find the values of $a$ and $b$ .

Set up two equations using the known terms:

\text{(1)}\quad 3.6 = 4a + b

\text{(2)}\quad 2.04 = 3.6a + b

Subtract (2) from (1):

1.56 = 0.4a \implies a = 3.9

Substitute $a=3.9$ into (1):

3.6 = 4(3.9) + b

3.6 = 15.6 + b \implies b = -12

So $a = 3.9$ and $b = -12$ .