Chapter 2 · Sequences · Concept 1
2-1 Sequences and General Terms
A trainer plans a recovery run for an injured athlete: 10 minutes a day in week one, then 3 more minutes every week — 13, then 16, then 19. To find week 24 you could write out every week. Or you could find the one rule that gives any week straight away.
Guiding question
How do you find the one rule that gives every term of a list?
Learning outcome. Find the general term for a sequence, and deduce some of its terms.
The words first
A list of numbers written in a row is a sequence. Each number in it is a term.
We name the terms by their position: a₁ is the first term, a₂ the second, and aₙ is the term in position n. The whole sequence is written {aₙ}.
Example
{aₙ} : 2, 4, 8, 16, …
a₁ = 2, a₂ = 4, a₃ = 8, a₄ = 16
When aₙ is written as an expression in n, that expression is called the general term. Put 1, 2, 3, … in place of n and the terms come out.
aₙ = −2n + 7 → a₁ = 5, a₂ = 3, a₃ = 1
The easy way
Formula → terms
You are given aₙ = 5n − 8. Put in n = 1, 2, 3 and work it out. There is nothing to think about.
a₁ = 5(1) − 8 = −3
The one to practise
Terms → formula
You are given 1, 2/3, 3/5, 4/7, … and you must find the rule. This feels impossible until you stop looking at the whole number.
A term like −3/5 is really three little things stuck together: a sign, a top and a bottom. Each one on its own follows an easy pattern. Solve them separately and the hard question falls apart.
Almost every piece you will meet is one of these. Learn them and most questions become reading.
2, 4, 6, 8, …
2n
the even numbers
1, 3, 5, 7, …
2n − 1
the odd numbers
+ − + − …
(−1)ⁿ⁻¹
flips, starting plus
The sign is the one people get backwards.
If the first term is plus, use (−1)ⁿ⁻¹. If the first term is minus, use (−1)ⁿ. Test it on n = 1 before you write anything else — it takes two seconds and saves the whole question.
1. The general term of {aₙ} is nn + 1. Find the first three terms.
- 1
Put n = 1 in, everywhere you see an n.
a₁ = 1/(1 + 1)Both the top and the bottom have an n in them. Change both.
2. Deduce the general term of each sequence.
Work with a partner. For question 2, say out loud which pieces you are splitting the sequence into before either of you writes a formula.
1. Find the first three terms.
2. Deduce the general term.
1. Find the first three terms.
2. Deduce the general term.
1 × 12, 2 × 13, 3 × 14, 4 × 15, 5 × 16, …
Deduce the general term, then find the 20th term.
Someone looks at 2, 4, 8, 16, … and says the rule is aₙ = 2n. Their first two terms are right. How would you show them they are wrong, and what does that tell you about testing formulas?
Try their formula on the third term.
Getting the sign power backwards.
First term plus means (−1)ⁿ⁻¹. First term minus means (−1)ⁿ. Put n = 1 in and check before writing anything else.
Forgetting the "one behind" sequences.
0, 1, 4, 9 is not n² — that would start at 1. It is (n − 1)². Always check what term 1 should give.
Trying to see the whole term at once.
Split it: sign, top, bottom. Three easy questions beat one impossible one.
Changing only some of the n's.
In n/(n + 1) there are two n's. Both become 3 when n = 3, giving 3/4 — not 3/(n + 1) or n/4.
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