Step by step
The question
Question: find the sum 1 + 3 + 5 + and so on + 39.
1 Find the difference
2 Count the terms
Careful!
(39 − 1)/2 counts the jumps.
There is one more term than jumps.
The rule from the sequences lesson: aₙ = 2n − 1.
The last term, 39, is term number n.
So we solve 2n − 1 = 39.
So 2n − 1 = 39.
We want n by itself.
Two things are stuck to n: times 2 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 2.
To remove − 1, we add 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.
39 + 1 = 40.
Now we have 2n = 40.
2n means 2 times n.
We want just n.
To remove times 2, we divide by 2.
We do it on both sides, so they stay equal.
2 divided by 2 is 1.
So just n is left.
40 divided by 2 is 20.
So n = 20!
So there are 20 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 = 20, the first term 1 and the last term 39.
1 + 39 is 40.
20 times 40 is 800.
Half of that is 400.
4 Check
Let's check the pairs: 1 + 39 is 40.
The second and the second-last: 3 + 37 is 40 too.
20 terms make 10 pairs.
10 times 40 is 400.
It works!
Let's look back.
First: find the difference.
Then: count the terms.
Then: add in pairs.
Last: check.