sum of all integers from 132 to 531 inclusive

[RonPurewal]

from a student:

What is the sum of all integers from 132 to 531, inclusive?

--

in this problem, you're looking for the SUM of a large SET OF NUMBERS.

when it comes to large SUMS, you should use the SUM FORMULA:
SUM = AVERAGE x NUMBER OF DATA POINTS

in this case, the NUMBER OF DATA POINTS is 400 (i.e., there are 500 integers being summed).
there are two ways to see this:
(1) use the "add one before you're done" rule (see the number properties guide):
number of integers = 531 - 132 + 1 = 532 - 132 = 400.
(2) if you subtract 131 from all of them, then 132, 133, 134, ..., 531 becomes 1, 2, 3, ..., 400. therefore, there are 400 integers. (this "matching technique" is only used for finding the NUMBER of integers in the list; obviously you can't do this when it comes to finding the average, or the sum, of the numbers.)

also, the AVERAGE is 331.5.
since this is a list of consecutive integers, you can just take the average of the first and last numbers, and that's the same as the average of the entire list.
this average is (132 + 531) / 2, or 331.5.

therefore, the sum is
400 x 331.5
= 132,600

[vishalsongra]

Thanks a lot ... its very useful short cut
:)

[cfaking]

Its AP with first term of 132 and last of 531 with d=1

now n=(531-132)+1=400

Sn=0.5(n)(first+last)=200*663

[Ben Ku]

Thanks cfaking. The formula for arithmetic series is another approach, if you know the formula. It's more helpful, though, for sequences where terms are not consecutive (i.e. d=1). Ultimately the Average formula that Ron used first is derived from the Arithmetic Series formula.