Step by step
The question
Question: an arithmetic sequence has a₃ = −4 and a₆ = 5.
Find the rule for term n.
1 Count the jumps
Term 3 is −4.
Term 6 is 5.
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 3: 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 5.
So −4 + 3d = 5.
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 add 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.
5 + 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
a₁ is 2 jumps before a₃.
Going back, we take away d each time.
2 times 3 is 6.
−4 take away 6 is −10.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = −10 and d = 3.
The bracket is times 3.
n times 3 is 3n.
−1 times 3 is −3.
Plain numbers go together: −10 − 3 is −13.
So the rule is aₙ = 3n − 13.
5 Check
Let's check with the terms we were given.
Put n = 3.
We get −4.
And n = 6 gives 5.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.