Step by step
The question
Question: an arithmetic sequence has a₄ = 30 and a₉ = 0.
Find the rule for term n.
1 Count the jumps
Term 4 is 30.
Term 9 is 0.
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 4: 5 jumps.
Careful!
We count the jumps, not the terms.
6 terms have only 5 jumps between them.
2 Find d
Start at 30, add d 5 times, and we get 0.
So 30 + 5d = 0.
We want d by itself.
Two things are stuck to d: times 5 and + 30.
The + 30 is on the outside, so it comes off first.
Then the times 5.
To remove 30, we take away 30.
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.
30 and − 30 make zero.
Cross them out.
0 − 30 = −30.
Now we have 5d = −30.
5d means 5 times d.
We want just d.
To remove times 5, we divide by 5.
We do it on both sides, so they stay equal.
5 divided by 5 is 1.
So just d is left.
−30 divided by 5 is −6.
So d = −6!
3 Find the first term
a₁ is 3 jumps before a₄.
Going back, we take away d each time.
3 times −6 is −18.
30 take away (−18) is 48.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 48 and d = −6.
The bracket is times −6.
n times −6 is −6n.
−1 times −6 is 6.
Plain numbers go together: 48 + 6 is 54.
So the rule is aₙ = −6n + 54.
5 Check
Let's check with the terms we were given.
Put n = 4.
We get 30.
And n = 9 gives 0.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.