Chapter 2 · Sequences · Concept 4
2-18 Difference Sequences
On day one, two people are infected. Then five, then ten, then seventeen. The virus is not spreading at a steady rate — but the way the rate itself changes follows a pattern, and finding that hidden pattern is how epidemiologists predict the peak.
Guiding question
What do you do when a sequence is neither arithmetic nor geometric?
Learning outcome. Find the general term of a sequence using its difference sequence.
Look one level down
The sequence made from the gaps between neighbouring terms is called the difference sequence.
10, 8, 4, −2, −10, …
its gaps: −2, −4, −6, −8, …
The original is neither arithmetic nor geometric. The gaps are plainly arithmetic.
when n ≥ 2
aₙ = a₁ + Σ (k = 1 to n − 1) bₖ
the first term, plus all the gaps you walk through to reach term n
Then check at n = 1, exactly as in lesson 2-17. If the formula gives back a₁, one line covers everything.
Pick a sequence you could not have handled last week. Write its gaps in a row below it, and see what they turn out to be.
Warm Up 1
Why it is n − 1 gaps, again.
Six terms have five gaps between them. You are already standing on term 1 for free, so reaching term n costs n − 1 steps — the same counting as lesson 2-2, one level down.
The Σ runs to n − 1, not n. So every n inside the summation formula has to become n − 1 — and the book's Common Mistakes panel is entirely about people who change some of them and not others.
right
Σ(k = 1 to n−1) of (4k − 1)
= 4 × ½(n−1){(n−1) + 1} − (n−1)
Every single n became n − 1, including the one that was on its own at the end.
wrong
Σ(k = 1 to n−1) of (4k − 1)
= 4 × ½n(n + 1) − n
That is the sum to n terms, not n − 1. It gives 2n² + n + 2 instead of 2n² − 3n + 3.
1. Find the general term of 2, 5, 12, 23, 38, 57, …
- 1
Check it is not arithmetic or geometric first.
gaps 3, 7, 11, … not equal · ratios 2.5, 2.4, … not equalIf either worked you would not need this lesson. Ten seconds to rule them out.
Work with a partner. Before either of you writes anything, both say whether the gaps are arithmetic or geometric. Everything after that follows from the answer.
Find the general term of each sequence.
A note on part (c).
The textbook prints 2, 3, 1, 7, −3, 13. The gaps of that are 1, −2, 6, −10, 16 — which is neither arithmetic nor geometric, so the method cannot start.
The sequence it was almost certainly meant to be is 2, 3, 1, 5, −3, 13, whose gaps are 1, −2, 4, −8, 16 — geometric with ratio −2. Five of the six printed terms already agree with that reading. Working it through gives aₙ = (7 − (−2)ⁿ⁻¹)/3.
Worth doing anyway, because spotting that a sequence does not fit is itself the skill. If the gaps are neither kind, stop and check the numbers before hunting for a clever method.
A tissue mass is measured over six weeks. Its volume in mm3 goes 1, 3, 7, 13, 21, 31, …. Find the general formula. If the volume passes 130 mm3 by week 12, surgery is needed. Work out a₁₂ and decide.
Every sequence in this lesson had gaps that were arithmetic or geometric. What would you do if the gaps were neither — and what does that suggest about how far this idea goes?
You are allowed to use the method more than once.
Summing to n instead of n − 1.
Six terms have five gaps. Reaching term n costs n − 1 of them, so the Σ stops at n − 1.
Changing only some of the n's.
In 4 × ½n(n+1) − n every n becomes (n − 1). The textbook devotes its whole Common Mistakes panel to this one.
Not naming the difference sequence.
Write "let {bₙ} be the difference sequence" and "when n ≥ 2". Both are marked.
Skipping the n = 1 check.
The formula is only proved for n ≥ 2. Substituting n = 1 and comparing with a₁ is the last required step.
Assuming the gaps must be arithmetic.
They are geometric just as often. Work out two gaps and two ratios before choosing.
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