Step by step
The question
Question: an arithmetic sequence has a₄ = 20 and a₈ = 36.
Find the rule for term n.
1 Count the jumps
Term 4 is 20.
Term 8 is 36.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
8 take away 4: 4 jumps.
Careful!
We count the jumps, not the terms.
5 terms have only 4 jumps between them.
2 Find d
Start at 20, add d 4 times, and we get 36.
So 20 + 4d = 36.
We want d by itself.
Two things are stuck to d: times 4 and + 20.
The + 20 is on the outside, so it comes off first.
Then the times 4.
To remove 20, we take away 20.
We do it on both sides.
Why both sides?
The = sign means both sides are the same amount.
If we change only one side, they are not the same any more.
20 and − 20 make zero.
Cross them out.
36 − 20 = 16.
Now we have 4d = 16.
4d means 4 times d.
We want just d.
To remove times 4, we divide by 4.
We do it on both sides, so they stay equal.
4 divided by 4 is 1.
So just d is left.
16 divided by 4 is 4.
So d = 4!
3 Find the first term
a₁ is 3 jumps before a₄.
Going back, we take away d each time.
3 times 4 is 12.
20 take away 12 is 8.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 8 and d = 4.
The bracket is times 4.
n times 4 is 4n.
−1 times 4 is −4.
Plain numbers go together: 8 − 4 is 4.
So the rule is aₙ = 4n + 4.
5 Check
Let's check with the terms we were given.
Put n = 4.
We get 20.
And n = 8 gives 36.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.