Step by step
The question
Question: an arithmetic sequence has a₁ = 4 and a₄ = 13.
Find the rule for term n.
1 Count the jumps
Term 1 is 4.
Term 4 is 13.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
4 take away 1: 3 jumps.
Careful!
We count the jumps, not the terms.
4 terms have only 3 jumps between them.
2 Find d
Start at 4, add d 3 times, and we get 13.
So 4 + 3d = 13.
We want d by itself.
Two things are stuck to d: times 3 and + 4.
The + 4 is on the outside, so it comes off first.
Then the times 3.
To remove 4, we take away 4.
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.
4 and − 4 make zero.
Cross them out.
13 − 4 = 9.
Now we have 3d = 9.
3d means 3 times d.
We want just d.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just d is left.
9 divided by 3 is 3.
So d = 3!
3 Find the first term
Term 1 is given: 4.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 4 and d = 3.
The bracket is times 3.
n times 3 is 3n.
−1 times 3 is −3.
Plain numbers go together: 4 − 3 is 1.
So the rule is aₙ = 3n + 1.
5 Check
Let's check with the terms we were given.
Put n = 1.
We get 4.
And n = 4 gives 13.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.