Chapter 2 · Sequences · Concept 3
2-16 Sequence where the General Term is a Sum
Look at 3, then 3 + 5, then 3 + 5 + 7. Each term is not a number you are handed — it is itself a little sum that grows by one piece every time. So finding the total of the first n terms means summing twice, one sum inside the other.
Guiding question
What if each term of a sequence is itself a sum?
Learning outcome. Deduce the sum of a sequence when its general term is expressed as the sum of some of its terms.
Do the inside first
When each term aₖ is itself a sum, work out aₖ as a formula in k before you go anywhere near the total.
Step 1 — the inner sum, over i
aₖ = Σ (i = 1 to k) of (the pattern)
Step 2 — the outer sum, over k
Sₙ = Σ (k = 1 to n) of aₖ
Use two different letters.
The inner one counts inside a single term, so it stops at k. The outer one counts which term you are on, so it stops at n. Using k for both makes the working say nothing, and it is the commonest mistake on these two pages.
Lay the sequence out and it makes a triangle. One row is one term. The whole triangle is the total. Two different sums, and now you can see both.
For the sequence 3, 3+5, 3+5+7, 3+5+7+9, … find (a) the k-th term and (b) the sum to n terms.
- 1
Look inside ONE term and find its pattern.
3, 5, 7, 9, … goes up in twos, so the pieces are 2i + 1Use i for the inner counting. At i = 1 you want 3, and 2 + 1 = 3 ✓
Work with a partner. One of you does the inner sum and the other does the outer one. Agree on which letter each of you is using before you start.
Find the general term of the sequence {aₙ} whose sum of the first n terms is Sₙ = 2n2 + 5n.
In Try 1, aₖ came out as exactly k². Look at the triangle for that sequence — can you see why the odd numbers add to a perfect square?
Think about building a square one L-shaped layer at a time.
Using k for both sums.
The inner sum runs over i up to k; the outer runs over k up to n. Using one letter for both makes the working meaningless.
Stopping after finding aₖ.
That is only part (a). The total needs a second summation, and it is worth as many marks.
Getting the number of pieces wrong.
The k-th term has k pieces, not k + 1. Count them on the first three terms before writing the Σ.
Not testing aₖ before using it.
Put k = 2 into your formula and compare with the second term written out. Ten seconds, and it catches almost everything.
Forgetting to factorise the answer at the end.
⅙n(n+1)(2n+7) is finished. Expanding it into a cubic is the same number and fewer marks.
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