SOLUTION: How many terms of the arithmetic progression 24,22,20 are needed to give the sum of 150?

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Question 796191: How many terms of the arithmetic progression 24,22,20 are needed to give the sum of 150?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
THE MENTAL MATH WAY:
The sum of an arithmetic progression is the average of the first and last terms times the number of therms.
If we add all the numbers between 24=2%2A12 and 2=2%2A1, we would be adding 12 terms, and multiplying times 12 is not easy.
Leaving out 2, and 4, I would have just 10 terms, and
24+22+20+ ,,, +10+8+6 =%28%2824%2B6%29%2F2%29%2A10=15%2A10=150
So adding highlight%2810%29 we get 150.

WITH FORMULAS
That progression has a first term
a%5B1%5D=24
and a common difference d=22-24=-2
The sum of the first n terms of an arithmetic progression can be calculated as
%28n%2F2%29%282a%5B1%5D%2B%28n-1%29%2Ad%29
In this case that sum would be 150, so
%28n%2F2%29%282%2A24%2B%28n-1%29%2A%28-2%29%29=150
%28n%2F2%29%2848-2n%2B2%29=150
%28n%2F2%29%2850-2n%29=150
n%2850-2n%29=150%2A2
50n-2n%5E2=300
2n%5E2-50n%2B300=0
n%5E2-25n%2B150=0
%28n-15%29%28n-10%29=0 --> system%28n=10%2C%22or%22%2Cn=15%29
Adding the 10 terms 24+22+ ... +10+8+6,
or adding the 15 terms 24+22+ ... +10+8+6 +4+2+0-2-4
we get 150,
but I would say highlight%2810%29 terms are needed to get to 150.
There is no need to keep adding 4+2+0+(-2)+(-4)=0.