Chapter 2 · Sequences · Concept 1
2-3 Arithmetic Sequence (2)
Last lesson you were handed the first term and the difference on a plate. Real questions are not that kind. They tell you the 3rd term and the 8th term and expect you to work backwards to the sequence itself.
Guiding question
If you only know two terms, can you rebuild the whole sequence?
Learning outcome. Calculate the general term of an arithmetic sequence, and find some specific terms in it.
Everything in terms of a and d
The general term is still aₙ = a + (n − 1)d.
To find the general term, you must first find a and d. Any two facts about the sequence give you two equations, and two equations give you two unknowns.
and one small fact worth memorising
if a, b, c are arithmetic in that order, then
2b = a + c
Why? The middle one is b = a + d and the last is c = a + 2d. So a + c = 2a + 2d, which is exactly 2(a + d) = 2b. In words: the middle number is the average of its two neighbours.
Write both facts, then take one away from the other. The a cancels because it is in both — and suddenly you have d on its own. That subtraction is the move worth learning.
Warm Up 1
- press the button to start
1. The 3rd term is 10 and the 8th term is 50. Find the general term.
- 1
Say what you are using.
let aₙ = a + (n − 1)dWrite this line every time. It costs nothing and markers look for it.
2. The common difference is 4 and the 8th term is 33.
3. x, 6, x2 form an arithmetic sequence in this order. Find x.
“Find the first term bigger than 500” ends with something like n > 124.75. There is no term number 124.75, so you have to decide which whole number to take — and the safe way is to check both.
Warm Up 2(b) — find the first term that is bigger than 500.
4n + 1 > 500 → n > 124.75
Solve it like an ordinary inequality. The answer comes out as a plain number, often a decimal — and a term number cannot be a decimal.
term 124
497
not past 500 yet ✗
term 125
501
past 500 ✓ — this is the answer
The answer is 125, not 124. When n must be bigger than 124.75, the smallest whole number that works is 125. Never round to nearest — ask which whole number is the first one past the line, and check the one before it fails.
Work with a partner. For question 2, one of you gives the term number and the other gives its value — then check you are not both saying the same thing.
1. Find the general term.
The four numbers 2, x, y, z form an arithmetic sequence in this order, and 2x, y + 1, z − y also form an arithmetic sequence in this order. Find x, y and z.
Exercise 3(b) asks for the first term smaller than −500, and the answer is the 171st term. But term 170 is exactly −500. Why is that not the answer, and what would change if the question said 'smaller than or equal to −500'?
Read the word 'smaller' very carefully.
Answering with the value instead of the position.
"Which term is 113?" wants 28. Writing 113 answers a question nobody asked. Read the word "which".
Rounding a threshold to the nearest whole number.
n > 124.75 gives 125 — but n > 34 gives 35, not 34. Do not round; ask which whole number is the first one past the line, and check the one before it fails.
Using n instead of (n − 1) in the equations.
The 8th term is a + 7d. Every wrong answer in this lesson can be traced back to that one slip.
Giving only one root.
2b = a + c often ends in a quadratic. Two roots means two answers, and both are worth marks unless the question rules one out.
Never checking d in the second equation.
You used one fact to find a. The other fact is a free check — spend the ten seconds.
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