Step by step
The question
Question: find the sum 50 + 46 + 42 + and so on + 2.
1 Find the difference
2 Count the terms
Careful!
(2 − 50)/4 counts the jumps.
There is one more term than jumps.
The rule from the sequences lesson: aₙ = −4n + 54.
The last term, 2, is term number n.
So we solve −4n + 54 = 2.
So −4n + 54 = 2.
We want n by itself.
Two things are stuck to n: times −4 and + 54.
The + 54 is on the outside, so it comes off first.
Then the times −4.
To remove + 54, we take away 54.
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.
+ 54 and − 54 make zero.
Cross them out.
2 − 54 = −52.
Now we have −4n = −52.
−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.
−52 divided by −4 is 13.
So n = 13!
So there are 13 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 = 13, the first term 50 and the last term 2.
50 + 2 is 52.
13 times 52 is 676.
Half of that is 338.
4 Check
Let's check the pairs: 50 + 2 is 52.
The second and the second-last: 46 + 6 is 52 too.
13 terms make 6 pairs and one term in the middle, 26.
6 times 52, plus 26, is 338.
It works!
Let's look back.
First: find the difference.
Then: count the terms.
Then: add in pairs.
Last: check.