sum of integers questions

gnod wrote:hi everyone,

This isn't an OG question but as I was reviewing my index cards a thought occurred to me and I wanted to make sure about it.

Say for instance we're asked to find the Sum of all multiples of 4 between 0 and 100.

I know one method is to find the (average) * (number of multiples between 0-100) = total sum

But is the average (0+100)/2? OR is it (4+100)/2? It's the former, correct? Since 0 is a multiple of every number?

Furthermore, the number of multiples is ((100-0)/4)+1 right? I'm getting myself a little confused as this thought never occurred to me.

It's hard to answer your question because people might interpret "multiples of 4 between 0 and 100" differently.
If you mean the sum of the multiples of 4 from 0 from 100 inclusive, then the following is true:
The NUMBER of terms = ((100-0)/4) + 1 = 26
The SUM of the terms = [(0+100)/2][26] = 1300

Cheers,
Brent

If you're not sure about certain sums, you can always rewrite the sum as something more familiar and easier to analyze.
Take the sum 0 + 4 + 8 + 12 + 16 + ..... + 96 + 100
We can rewrite this as 4(0 + 1 + 2 + 3 + 4 + .... + 24 + 25)
At this point, it's easy to see that there are 26 terms altogether.
The average of the values from 0 to 25 inclusive = (0 + 25)/2 = 12.5

So, the TOTAL SUM = (4)(26)(12.5) = 1300

Cheers,
Brent

A super fast way to find 0 + 4 + 8 + 12 + 16 + ..... + 96 + 100 is to add the terms in PAIRS starting from the OUTSIDE and work our way in.

We get: (0 + 100) + (4 + 96) + (8 + 192) + (12 + 188) + ....
Notice that each pair of values add to 100.
How many pairs are we adding altogether?

Well, in an earlier post, we learned that there are 26 terms altogether. So, there are 13 PAIRS to add.
Since each pair of values add to 100, the TOTAL SUM = (13)(100) = 1300

Cheers,
Brent

Hi gnod,

In your original post, you showed that you understand the "algebra" approach to solving that sum. You'll get the same answer in either scenario, as long as you're paying attention to the details.

For the multiples of 4 from 0 to 100, inclusive, if you include the number 0, then you have 26 terms.

The average of the smallest and the biggest is (0 + 100)/2 = 50

50(26) = 1300

For the multiples of 4 from 4 to 100, inclusive, then you DON'T have the 0, so you have 25 terms.

The average of the smallest and the biggest is (4 + 100)/2 = 52

52(25) = 1300

So, either way you think about it, you should get the correct answer. You just have to make sure that you adjust your math according to the "terms" you're adding up.

GMAT assassins aren't born, they're made,
Rich