How do i solve the following summation problem?

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1 Answer
May 9, 2018

There are few enough terms to just do it.


sum_(n=4)^(11) (30 - 4n)

= (30 - 4(4)) + (30 - 4(5)) + (30 - 4(6)) + (30 - 4(7)) + (30 - 4(8)) + (30 - 4(9)) + (30 - 4(10)) + (30 - 4(11))

From term 4 through 11 inclusive, there are 8 values of 30, so:

= 8(30) - 4(4) - . . . - 4(11)

= 8(30) - 4(4 + . . . + 11)

We know that 1 + . . . + 11 = (11*12)/2 = 66, so

4 + . . . + 11 = 1 + . . . + 11 - (1 + 2 + 3)

= 66 - 1 - 2 - 3 = 60.

Therefore, 4(4 + . . . + 11) = 240, and:

sum_(n=4)^(11) (30 - 4n) = 240 - 240 = color(blue)(0)