Step by step
The question
Question: an arithmetic sequence has a₃ = 10 and a₇ = 22.
Find the rule for term n.
1 Count the jumps
Term 3 is 10.
Term 7 is 22.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
7 take away 3: 4 jumps.
Careful!
We count the jumps, not the terms.
5 terms have only 4 jumps between them.
2 Find d
Start at 10, add d 4 times, and we get 22.
So 10 + 4d = 22.
We want d by itself.
Two things are stuck to d: times 4 and + 10.
The + 10 is on the outside, so it comes off first.
Then the times 4.
To remove 10, we take away 10.
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.
10 and − 10 make zero.
Cross them out.
22 − 10 = 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 2 jumps before a₃.
Going back, we take away d each time.
2 times 3 is 6.
10 take away 6 is 4.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 4 and d = 3.
The bracket is times 3.
n times 3 is 3n.
−1 times 3 is −3.
Plain numbers go together: 4 − 3 is 1.
So the rule is aₙ = 3n + 1.
5 Check
Let's check with the terms we were given.
Put n = 3.
We get 10.
And n = 7 gives 22.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.