Step by step
The question
Question: find the sum 5 + 9 + 13 + and so on + 85.
1 Find the difference
2 Count the terms
Careful!
(85 − 5)/4 counts the jumps.
There is one more term than jumps.
The rule from the sequences lesson: aₙ = 4n + 1.
The last term, 85, is term number n.
So we solve 4n + 1 = 85.
So 4n + 1 = 85.
We want n by itself.
Two things are stuck to n: times 4 and + 1.
The + 1 is on the outside, so it comes off first.
Then the times 4.
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.
85 − 1 = 84.
Now we have 4n = 84.
4n means 4 times n.
We want just n.
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 n is left.
84 divided by 4 is 21.
So n = 21!
So there are 21 terms.
3 Add in pairs
The pairs rule from the sequences lesson: first plus last makes every pair.
Times the number of terms, over 2.
Put in n = 21, the first term 5 and the last term 85.
5 + 85 is 90.
21 times 90 is 1890.
Half of that is 945.
4 Check
Let's check the pairs: 5 + 85 is 90.
The second and the second-last: 9 + 81 is 90 too.
21 terms make 10 pairs and one term in the middle, 45.
10 times 90, plus 45, is 945.
It works!
Let's look back.
First: find the difference.
Then: count the terms.
Then: add in pairs.
Last: check.