Step by step
The question
Question: an arithmetic sequence has a₂ = 15 and a₆ = 3.
Find the rule for term n.
1 Count the jumps
Term 2 is 15.
Term 6 is 3.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
6 take away 2: 4 jumps.
Careful!
We count the jumps, not the terms.
5 terms have only 4 jumps between them.
2 Find d
Start at 15, add d 4 times, and we get 3.
So 15 + 4d = 3.
We want d by itself.
Two things are stuck to d: times 4 and + 15.
The + 15 is on the outside, so it comes off first.
Then the times 4.
To remove 15, we take away 15.
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.
15 and − 15 make zero.
Cross them out.
3 − 15 = −12.
Now we have 4d = −12.
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.
−12 divided by 4 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.
15 take away (−3) is 18.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 18 and d = −3.
The bracket is times −3.
n times −3 is −3n.
−1 times −3 is 3.
Plain numbers go together: 18 + 3 is 21.
So the rule is aₙ = −3n + 21.
5 Check
Let's check with the terms we were given.
Put n = 2.
We get 15.
And n = 6 gives 3.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.