Step by step
The question
Question: an arithmetic sequence has a₆ = 0 and a₁₀ = 8.
Find the rule for term n.
1 Count the jumps
Term 6 is 0.
Term 10 is 8.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
10 take away 6: 4 jumps.
Careful!
We count the jumps, not the terms.
5 terms have only 4 jumps between them.
2 Find d
Start at 0, add d 4 times, and we get 8.
So 0 + 4d = 8.
We want d by itself.
Two things are stuck to d: times 4 and + 0.
The + 0 is on the outside, so it comes off first.
Then the times 4.
To remove + 0, we add 0.
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.
+ 0 and + 0 make zero.
Cross them out.
8 + 0 = 8.
Now we have 4d = 8.
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.
8 divided by 4 is 2.
So d = 2!
3 Find the first term
a₁ is 5 jumps before a₆.
Going back, we take away d each time.
5 times 2 is 10.
0 take away 10 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 = 2.
The bracket is times 2.
n times 2 is 2n.
−1 times 2 is −2.
Plain numbers go together: −10 − 2 is −12.
So the rule is aₙ = 2n − 12.
5 Check
Let's check with the terms we were given.
Put n = 6.
We get 0.
And n = 10 gives 8.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.