Step by step
The question
Question: find the sum −7 − 4 − 1 + and so on + 32.
1 Find the difference
2 Count the terms
Careful!
(32 − (−7)) over 3 counts the jumps.
There is one more term than jumps.
The rule from the sequences lesson: aₙ = 3n − 10.
The last term, 32, is term number n.
So we solve 3n − 10 = 32.
So 3n − 10 = 32.
We want n by itself.
Two things are stuck to n: times 3 and − 10.
The − 10 is on the outside, so it comes off first.
Then the times 3.
To remove − 10, we add 10.
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.
− 10 and + 10 make zero.
Cross them out.
32 + 10 = 42.
Now we have 3n = 42.
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.
42 divided by 3 is 14.
So n = 14!
So there are 14 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 = 14, the first term −7 and the last term 32.
−7 + 32 is 25.
14 times 25 is 350.
Half of that is 175.
4 Check
Let's check the pairs: −7 + 32 is 25.
The second and the second-last: −4 + 29 is 25 too.
14 terms make 7 pairs.
7 times 25 is 175.
It works!
Let's look back.
First: find the difference.
Then: count the terms.
Then: add in pairs.
Last: check.