Step by step
The question
Question: an arithmetic sequence has a₂ = 1 and a₇ = 11.
Find the rule for term n.
1 Count the jumps
Term 2 is 1.
Term 7 is 11.
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 2: 5 jumps.
Careful!
We count the jumps, not the terms.
6 terms have only 5 jumps between them.
2 Find d
Start at 1, add d 5 times, and we get 11.
So 1 + 5d = 11.
We want d by itself.
Two things are stuck to d: times 5 and + 1.
The + 1 is on the outside, so it comes off first.
Then the times 5.
To remove 1, we take away 1.
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.
1 and − 1 make zero.
Cross them out.
11 − 1 = 10.
Now we have 5d = 10.
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.
10 divided by 5 is 2.
So d = 2!
3 Find the first term
a₁ is 1 jump before a₂.
Going back, we take away d each time.
1 times 2 is 2.
1 take away 2 is −1.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = −1 and d = 2.
The bracket is times 2.
n times 2 is 2n.
−1 times 2 is −2.
Plain numbers go together: −1 − 2 is −3.
So the rule is aₙ = 2n − 3.
5 Check
Let's check with the terms we were given.
Put n = 2.
We get 1.
And n = 7 gives 11.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.