First posted as a hint, but now it is the compilation to @Atul Mishra's answer.

Let us see what is happening in diagrammatic way:

Now, look at the image carefully.The coloured regions represent following data:

*The rectangle: Sum of All the numbers from $1-100$.*

*White region: Sum of Numbers that are neither multiple of $3$ nor of $7$.*

*Green region: Sum of Numbers that are multiple of $3$ only.*

*Blue region: Sum of Numbers that are multiple of $7$ only.*

*Light blue region: Sum of Numbers that are multiple of $3$ and $7$.*

We have to find the sum of numbers which are neither divisible by $3$ nor by $7$. So, we will find the sum of numbers that are in white region.

Sum of Numbers in white region $=$ Sum of Numbers in Rectangle $-$ Sum of Numbers in (Dark blue $+$ Light Blue) $-$ Sum of Numbers in (Green + Light Blue) $+$ Sum of Numbers in light blue region.

From here we get the formula used by other answers. i.e.,

Sum of Required numbers $=$ Sum of Total Numbers $-$ Sum of Numbers divisible by $7-$ Sum of Numbers divisible by $3+$ Sum of Numbers divisible by both $3$ and $7$.

Next question you may ask is that, How to find the sum of numbers from $1-100$ or sum of multiples of $3$ etc.

There is no problem in it, you just have to identify the A.P.

Sum of numbers from $1$ to $100$ equals $\frac{100}{2}\times {101}$.

Sum of multiples of $3$ equals $\frac{33}{2}\times 102$.
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I think I shall let you conclude now. :) :) :)