SOLUTION: What is the sum of all the multiples of 4 through 4000?

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Question 723119: What is the sum of all the multiples of 4 through 4000?
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
That is the sum of an arithmetic sequence with 1000 terms.
It is easy to see that the sum,
4+8+12+ .... +3992+3996+4000 is half of
(4+8+12+ .... +3992+3996+4000) + (4000+3996+3992+ .... +12+8+4) = (4+4000)+(8+3996)+(12+3992)+ ... +(3992+12)+(3996+8)+(4000+4)=4004+4004+4004+ ... +4004+4004+4004=1000*4004=4004000
So the sum is highlight%282002000%29 .

We could write some fancy formulas, like
a%5B1%5D=4, d=4, a%5Bk%5D%2B4=a%5Bk%2B1%5D,
a%5Bk%5D=a%5B1%5D%2Bd%28k-1%29 and in this case a%5Bk%5D=4%2B4%28k-1%29=4%2B4k-4=4k
In this case, n=1000 a%5Bn%5D=4%2A1000=4000
sum%28a%5Bk%5D%2Ck=1%2C1000%29=n%2A%28a%5B1%5D%2Ba%5Bn%5D%29%2F2 and in this case
sum%284k%2Ck=1%2C1000%29=1000%2A%284%2B4000%29%2F2=1000%2A4004%2F2=2002000