Question

What is the sum of the multiples of 2 between 1 and 400?

Answer 1

40,200

Explanation:

Sum of the multiples of 2 between 1 and 400 is the sum of even integers from 2 to 400.

`= sum_(n=1)^200 2n`

This is an arithmetic sequence with first term `(a_1)=2` and last term `(a_200) = 400`

The sum of such a sequence is given by: `n/2(a_1+a_n)`

Hence `= sum_(n=1)^200 2n = 200/2(2+400)`

`= 100 xx 402 = 40,200`

Answer 2

`40200`

Explanation:

One way of looking at this is to rearrange the sum:

`2+4+6+...+396+398+400`

`=(400+2)+(398+4)+(396+6)+...+(202+200)`

`=overbrace(402+402+402+...+402)^"100 times"` `" "larr` half as many terms

`=40200`

The same rearrangement can be made with any arithmetic series.

So with any arithmetic series you can add together the first and last terms, multiply by the original number of terms and divide by `2` to get the sum.

That is, if `a_1, a_2,...,a_n` is an arithmetic sequence, then:

`a_1+a_2+...+a_n = n/2(a_1+a_n)`