Question

Find the sum of all the integers from 1 to 100 inclusive that are not
Multiples of 7.

Answer 1

Step-by-step explanation:

$\text{Denote The sum of all the integers from 1 to 100 inclusive that are NOT multiples of 7 as } S.$

$\text{Denote The sum of all the integers from 1 to 100 inclusive that are multiples of 7 as } P.$

$\text{Denote The sum of all the integers from 1 to 100 inclusive as } Q.$

Then

$\begin{align}S&=Q-P\\&=\frac{100(100+1)}{2}-(7+2\cdot7+\ldots+14\cdot7)\\&=\frac{100(100+1)}{2}-7\cdot\frac{14(14+1)}{2}\\&=4315\end{align}$