Addition of integers, looks easy!

Use PIE

Required sum $\begin{aligned} S &=\sum_{n=1}^{100}n-\left(\sum_{n=1}^{33}3n+\sum_{n=1}^{20}5n\right)+\sum_{n=1}^{6}15n\\ &=\dfrac{1}{2}(100 \cdot 101-3 \cdot 33 \cdot 34-5 \cdot 20 \cdot 21+15 \cdot 6 \cdot 7) \ \ \ \ \ \ \ \ \ \ \ \ \ \text{as} \ \sum_{k=1}^{n} k=\dfrac{n}{2}(n+1)\\ &=\boxed{2632} \end{aligned}$

We will first calculate the sum of the natural numbers which are divisible by 3 or 5 and then we will subtract it by the sum of the first 100 natural numbers. $CASE\ 1:$ The number is divisible by $3.$ The numbers are from $3*1---3*33\ as\ 3*34>100.$ $CASE\ 2:$ The number is divisible by $5.$ The numbers are from $5*1---5*20\ as\ 5*21>100.$ Sum of numbers of Case 1 $=3(1+2+...33$ $=3*\dfrac{33}{2}*34$ $=1683.$ Sum of numbers of Case 2 $=5*\dfrac{20}{2}*21$ $=1050.$ But we have counted some numbers twice which are the multiples of both $3$ and $5.$ We find them using this: $3*x=5*y.$ The solutions of this equation are $(5,3)(10,6)(15,9)(20,12)(25,15)(30,18).$ This is because the value of 'x' ranges from $1$ to $33.$ Thus,the numbers are: $15,30,45,60,75,90.$ Thus, the sum of the natural numbers which are divisible by 3 or 5 is: $1683+1050-15-30-45-60-75-90$ $=2418.$ Thus,our answer is sum of the first 100 natural numbers-2418= $\dfrac{100}{2}*101-2418$ $=\boxed{2632}$

The sum,

$S = 1+2+4+7+...+98$

$\quad = (1+2+3+...+100) - (3+6+9+...+99) - (5+10+15+...+100)+(15+30+45+60+75+90)$

$\quad = \sum _{n=1} ^{100} {n} - 3 \sum _{n=1} ^{33} {n} - 5 \sum _{n=1} ^{20} {n} + 15 \sum _{n=1} ^{6} {n}$

$\quad = \frac {100(100+1)}{2} - 3 \left( \frac {33(33+1)}{2} \right) - 5 \left( \frac {20(20+1)}{2} \right) + 15 \left( \frac {6(6+1)}{2} \right)$

$\quad = 5050 -1683 -1050 + 315 = \boxed{2632}$

I did it the hard way around. I added the numbers one by one(I couldn't resist the temptation of 55 points at once.) 1 + 2 + 4 + 7 + 8 + 11 + 13 + 14 + 16 + 17 + 19 + 22 + 23 + 26 + 28 + 29 + 31 + 32 + 34 + 35 + 2 + 38 + 41 + 43 + 44 + 46 + 47 + 49 + 51 - 51 + 52 + 53 + 56 + 58 + 59 + 61 + 62 + 64 + 67 + 68 + 71 + 73 + 74 + 76 + 77 + 79 + 82 + 83 + 86 + 88 + 89 + 91 + 92 + 94 + 97 + 98=2632 I know I don't deserve the points, but I got the answer.