Step by step
The question
Question: find the sum 2 + 5 + 8 + and so on + 47.
1 Find the difference
2 Count the terms
Careful!
(47 − 2)/3 counts the jumps.
There is one more term than jumps.
The rule from the sequences lesson: aₙ = 3n − 1.
The last term, 47, is term number n.
So we solve 3n − 1 = 47.
So 3n − 1 = 47.
We want n by itself.
Two things are stuck to n: times 3 and − 1.
The − 1 is on the outside, so it comes off first.
Then the times 3.
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.
47 + 1 = 48.
Now we have 3n = 48.
3n means 3 times n.
We want just n.
To remove times 3, we divide by 3.
We do it on both sides, so they stay equal.
3 divided by 3 is 1.
So just n is left.
48 divided by 3 is 16.
So n = 16!
So there are 16 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 = 16, the first term 2 and the last term 47.
2 + 47 is 49.
16 times 49 is 784.
Half of that is 392.
4 Check
Let's check the pairs: 2 + 47 is 49.
The second and the second-last: 5 + 44 is 49 too.
16 terms make 8 pairs.
8 times 49 is 392.
It works!
Let's look back.
First: find the difference.
Then: count the terms.
Then: add in pairs.
Last: check.