Chapter 2 · Sequences · Concept 5
2-22 Recurrence Relations (3)
aₙ₊₁ = 3aₙ is geometric and you can solve it in your head. aₙ₊₁ = 3aₙ + 4 has one extra number and suddenly nothing works. There is a trick, and it is much less mysterious than the five steps make it look.
Guiding question
How do you handle a rule that multiplies AND adds?
Learning outcome. Solve recurrence relations at different levels in order to deduce the terms of the sequence.
The five steps
For a rule of the shape aₙ₊₁ = p aₙ + q:
- 1
Replace both aₙ₊₁ and aₙ with the same letter c.
c = pc + q
- 2
Solve that little equation for c.
one number falls out
- 3
Rewrite the rule around c.
aₙ₊₁ − c = p(aₙ − c)
- 4
Let bₙ = aₙ − c. That is geometric with ratio p.
bₙ₊₁ = p bₙ
- 5
Find bₙ, then add c back to get aₙ.
aₙ = bₙ + c
And what c actually is
c is the value that stays put. If a term ever equalled c, the rule would hand back c again, and the sequence would sit there forever.
So measuring every term from c rather than from zero is the whole idea. The distance from c is what multiplies cleanly by p — the +q was only ever the offset between zero and c, and once you stop measuring from zero it disappears.
That is worth knowing rather than memorising, because if you forget step 3 you can rebuild it: “the gap from c multiplies by p” is aₙ₊₁ − c = p(aₙ − c).
Warm Up · a₁ = 1
| aₙ | 1 | 7 | 25 | 79 | 241 |
a₁ = 1 and aₙ₊₁ = 3aₙ + 4. Find the general term.
- 1
Replace both a's with c.
c = 3c + 4You are asking which value the rule leaves alone.
Work with a partner. One of you finds c and checks it stays put; the other does the geometric part. Then swap and check each other's first term.
Find the general term.
Parts (e) and (f) have a fraction for p. Nothing changes — c still comes from c = pc + q, and the shifted sequence is still geometric. The only difference is that its terms shrink instead of growing, so aₙ creeps towards c rather than running away from it.
In Exercise (e) the ratio is ⅓, so the shifted sequence shrinks towards zero. What does that mean for the original sequence — and what does it say about the antibiotic in the lesson opener?
If bₙ heads to zero, where does aₙ = bₙ + c head?
Using a₁ as the first term of the shifted sequence.
b₁ = a₁ − c, not a₁. In the Warm Up that is 3, not 1 — and every later term is wrong if you miss it.
Getting the sign wrong when c is negative.
With c = −2, the shifted sequence is aₙ − (−2) = aₙ + 2. Write the bracket in.
Forgetting to add c back at the end.
bₙ is not the answer. The last line must undo the shift: aₙ = bₙ + c.
Solving c = pc + q carelessly.
Check it before going on: put c back into the rule and confirm you get c again. Two seconds.
Treating aₙ₊₁ = 3aₙ + 4 as geometric.
The ratios of its terms are not equal — 7/1, 25/7, 79/25. Only the SHIFTED sequence is geometric.
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