Step by step
The question
Question: an arithmetic sequence has a₂ = 5 and a₅ = 14.
Find the rule for term n.
1 Count the jumps
Term 2 is 5.
Term 5 is 14.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
5 take away 2: 3 jumps.
Careful!
We count the jumps, not the terms.
4 terms have only 3 jumps between them.
2 Find d
Start at 5, add d 3 times, and we get 14.
So 5 + 3d = 14.
We want d by itself.
Two things are stuck to d: times 3 and + 5.
The + 5 is on the outside, so it comes off first.
Then the times 3.
To remove 5, we take away 5.
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.
5 and − 5 make zero.
Cross them out.
14 − 5 = 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
a₁ is 1 jump before a₂.
Going back, we take away d each time.
1 times 3 is 3.
5 take away 3 is 2.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 2 and d = 3.
The bracket is times 3.
n times 3 is 3n.
−1 times 3 is −3.
Plain numbers go together: 2 − 3 is −1.
So the rule is aₙ = 3n − 1.
5 Check
Let's check with the terms we were given.
Put n = 2.
We get 5.
And n = 5 gives 14.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.