Sums

For answers on this post highlight between the xxx's to see

Quickly tell me how much is

95+96+97=

Instead of doing the sum you could just approximate it
Like for example: 3*95

Or even better it could be...

xxx 3*96 xxx

But wait that is exactly 95+96+97

Since xxx 96+96+96 xxx = 95+96+97

As the terms even out.

Now how much is

94+95+96+97+98=

xxx 5*96! xxx

You can guess 93+94+95+96+97+98+99

But wait a second and if you have for example

94+95+96+97=?

There is no central one to average but still

The average should be xxx (95+96)/2 = 95.5 = (94+97)/2 xxx

So the sum is xxx 4*95.5 xxx

It is always the (number of terms)*(average or "middle term")

And now the last sum

37+39+...+79+81

In this case we don't see the middle term but you know what it should be?

xxx (37+81)/2 xxx

That is the (average) is (first+last)/2

So the sum is:

(number of terms)*(average)=(number of terms)*(first+last)/2

That is the sum of an arithmetic progression.