Step by step
The question
Question: an arithmetic sequence has a₅ = 7 and a₉ = −1.
Find the rule for term n.
1 Count the jumps
Term 5 is 7.
Term 9 is −1.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
9 take away 5: 4 jumps.
Careful!
We count the jumps, not the terms.
5 terms have only 4 jumps between them.
2 Find d
Start at 7, add d 4 times, and we get −1.
So 7 + 4d = −1.
We want d by itself.
Two things are stuck to d: times 4 and + 7.
The + 7 is on the outside, so it comes off first.
Then the times 4.
To remove 7, we take away 7.
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.
7 and − 7 make zero.
Cross them out.
−1 − 7 = −8.
Now we have 4d = −8.
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.
−8 divided by 4 is −2.
So d = −2!
3 Find the first term
a₁ is 4 jumps before a₅.
Going back, we take away d each time.
4 times −2 is −8.
7 take away (−8) is 15.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 15 and d = −2.
The bracket is times −2.
n times −2 is −2n.
−1 times −2 is 2.
Plain numbers go together: 15 + 2 is 17.
So the rule is aₙ = −2n + 17.
5 Check
Let's check with the terms we were given.
Put n = 5.
We get 7.
And n = 9 gives −1.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.