Worked example 1
Generate terms
u₀ = 10 and uₙ₊₁ = 0.8uₙ + 3. Find u₁ and u₂
- Use u₀ to find u₁: 0.8(10) + 3.
- Use u₁ to find u₂.
Answer: u₁ = 11 and u₂ = 11.8
Back to Higher Mathematics
Higher Mathematics
Recurrence relations
Generate terms from uₙ₊₁ rules and interpret steady-state behaviour.
Before you start
- Substitute values into formulae carefully.
- Use the previous term, not always the first term.
- Track rounding instructions.
Explanation
A recurrence relation defines each term using an earlier term. You need a starting value and a rule, such as uₙ₊₁ = auₙ + b.
Some recurrence relations approach a fixed point. A fixed point satisfies L = aL + b
Method and rules
- Use the given starting value first.
- uₙ₊₁ means the next term after uₙ.
- Fixed point: solve L = aL + b
Worked examples
Worked example 2
Find a fixed point
Find the fixed point of uₙ₊₁ = 0.5uₙ + 6.
- Set L = 0.5L + 6
- Subtract 0.5L from both sides.
- Solve for L.
So: L = 12
Watch out
- Using the wrong starting value.
- Putting n into the formula when the rule needs uₙ.
- Rounding too early.
- Treating a fixed point as the first term.
Exam reminder
Show at least the first substitution clearly. If a decimal sequence is involved, follow the rounding instruction at each stated stage.