Find the sum of all the integers between 3 and 205. Hence , calculate the sum of these integers that are not multiples of 7.

Answer 1

An important formula here is that the sum of integers from 1 to n is equal to n(n+1)/2.

So the sum of integers between 1 and 205 is 205(206) / 2 = 42,230 / 2 = 21,115.

Now let's subtract 1 + 2 =3 since we only want the sum of integers between 3 and 205. So 21,115 - 3 = 21,112.

Integers that are multiples of 7 will be 7, 14, 21, 28, ..., 203 = 7(1 + 2 + 3 + ... + 29), or 7 times the sum of integers between 1 and 29. Again using our formula above, the sum of integers from 1 to 29 is 29(30)/2 = 870 / 2 = 435. So 7(435) = 3045, and therefore the sum of integers between 3 and 205 that are multiples of 7 is 3045.

So if we want the sum of integers between 3 and 205 that are not multiples of 7, we calculate 21,112 - 3045 = 18,067.