Step by step
The question
Question: an arithmetic sequence has a₁₀ = 50 and a₁₅ = 75.
Find the rule for term n.
1 Count the jumps
Term 10 is 50.
Term 15 is 75.
We don't know the terms between.
In an arithmetic sequence we add the same number d every time.
How many jumps?
15 take away 10: 5 jumps.
Careful!
We count the jumps, not the terms.
6 terms have only 5 jumps between them.
2 Find d
Start at 50, add d 5 times, and we get 75.
So 50 + 5d = 75.
We want d by itself.
Two things are stuck to d: times 5 and + 50.
The + 50 is on the outside, so it comes off first.
Then the times 5.
To remove 50, we take away 50.
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.
50 and − 50 make zero.
Cross them out.
75 − 50 = 25.
Now we have 5d = 25.
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.
25 divided by 5 is 5.
So d = 5!
3 Find the first term
a₁ is 9 jumps before a₁₀.
Going back, we take away d each time.
9 times 5 is 45.
50 take away 45 is 5.
4 Write the rule
The rule from the sequences lesson: the first term, plus (n − 1) times the difference.
Put in a₁ = 5 and d = 5.
The bracket is times 5.
n times 5 is 5n.
−1 times 5 is −5.
Plain numbers go together: 5 − 5 is 0.
So the rule is aₙ = 5n.
5 Check
Let's check with the terms we were given.
Put n = 10.
We get 50.
And n = 15 gives 75.
Both work!
Let's look back.
First: count the jumps.
Then: find d.
Then: find the first term.
Then: write the rule.
Last: check.