Chapter 2 · Sequences · Concept 3
2-17 Sum of Sequence and General Term
Everything so far went one way: terms in, total out. This lesson runs it backwards. You are handed the running total and asked what the terms were — and there is a catch at the very first one that costs more marks than anything else in the chapter.
Guiding question
If you know the running total, can you get the terms back?
Learning outcome. Connect the sum of a sequence with its general term, and deduce one from the other.
A term is what the total gained
aₙ = Sₙ − Sₙ₋₁
but only when n ≥ 2
The total after n terms, take away the total after n − 1 terms, is exactly the term you added in between. And separately:
a₁ = S₁
Why n ≥ 2, and why it matters.
At n = 1 the formula asks for S₀ — the total of no terms at all. That is not something the question gave you, so the working is not allowed there.
So find a₁ from S₁ separately, then compare:
- · if they agree → keep the one formula, and say that it also holds at n = 1
- · if they disagree → write both: a₁ = …, and aₙ = … when n ≥ 2
Five sums, all handled the same way. Three of them turn out fine at n = 1 and two of them do not — and there is no way to tell in advance. Press the button and see.
Sₙ = n² − 3n → aₙ = Sₙ − Sₙ₋₁ = 2n − 4
This working is only allowed for n ≥ 2, because Sₙ₋₁ means nothing when n = 1.
what the formula gives at n = 1
?
what S₁ actually is
?
Nothing in the algebra warns you which way it will go. Try all five — two of them fail, and there is no pattern to spot. Only the check tells you.
(a) Find the general term when Sn = n2 − 3n.
- 1
Write the condition down before the working. It is worth a mark on its own.
when n ≥ 2:The book's margin says "Deduction if not written" four separate times. Write it.
Work with a partner. One of you does the n ≥ 2 working while the other works out S₁. Then compare — and one of these three does not agree.
Find the general term. Four of these eight need two lines — check every one at n = 1.
Sₙ = 3n+1 − 2n + 1
Find the general term.
Sₙ = 5ⁿ − 1 works at n = 1, but Sₙ = 5ⁿ − 2 does not. The two differ by a single constant. Why does changing that number decide the whole question?
Work out S₁ in each case, and compare with what the formula predicts.
Not writing "when n ≥ 2".
The textbook margin says "Deduction if not written" four times on one page. It is a free mark and it is easy to forget.
Skipping the n = 1 check.
Half the exercises on that page fail at n = 1. Working out S₁ and comparing is the difference between a right answer and a nearly-right one.
Missing an n when writing Sₙ₋₁.
Every single n becomes (n − 1). In 3ⁿ⁺¹ − 2n + 1 that is two separate places.
Forgetting the bracket when subtracting.
aₙ = Sₙ − (Sₙ₋₁). Without the bracket only the first term of Sₙ₋₁ gets its sign flipped.
Working powers out as numbers.
5ⁿ − 5ⁿ⁻¹ factorises to 4 × 5ⁿ⁻¹. Turning the powers into digits makes the pattern impossible to see.
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