Sorry, I posted this on page 1228, but nobody helped me on the second part. Sorry to bother you guys.

Part (a): Find the sum  s =   in terms of  and  

$$s = a + (a+1) + ( a+2) + (a+3) + ... +(a+ (n-2)) + (a+(n-1))\\\\ s = \left[a + (a+(n-1))\right] *(\frac{n}{2} ) \\\\ s = \left[2a+(n-1))\right]*(\frac{n}{2} ) \\\\ \boxed{s = n*a+\frac{n(n-1)}{2}}$$

Part (b): Find all pairs of positive integers  such that  and 

$$\small{\text{$ 2\le n\le14 \text{ and } a > 0 $}}\\ \small{\text{ $ n= 2\quad a=49.500000 $ }} \\ \small{\text{ $ n= 3\quad a=32.333333 $ }} $\\$ \small{\text{ $ n= 4\quad a=23.500000 $ }} $\\$ \small{\text{ $\textcolor[rgb]{1,0,0}{n= 5\quad a=18.000000} $ }} $\\$ \small{\text{ $ n= 6\quad a=14.166667 $ }} $\\$ \small{\text{ $ n= 7\quad a=11.285714 $ }} $\\$ \small{\text{ $ \textcolor[rgb]{1,0,0}{n= 8\quad a=9.000000 }$ }} $\\$ \small{\text{ $n= 9\quad a=7.111111 $ }} $\\$ \small{\text{ $n=10\quad a=5.500000 $ }} $\\$ \small{\text{ $n=11\quad a=4.090909 $ }} $\\$ \small{\text{ $n=12\quad a=2.833333 $ }} $\\$ \small{\text{ $n=13\quad a=1.692308 $ }} $\\$ \small{\text{ $n=14\quad a=0.642857 $ }} $\\$ \small{\text{ The only 2 solutions for $(a,n)$ are $ (18,5),\ (9,8)$ }} $\\$ \small{\text{ $ \textcolor[rgb]{1,0,0}{18}+19+20+21+22 = 100 \quad $ and $\quad \textcolor[rgb]{1,0,0}{9}+10+11+12+13+14+15+16 = 100 $ }}$$

http://web2.0calc.com/questions/instructions-on-reposting_1

It is best to follow these instructions when you want to repost :)

Thanks Heureka    

My answer is wrong - the error was right near the beginning.    

I am sure that Heureka's answer is perfect.

Heureka, could you please explain how you got $2\leq n\leq14$ and $a>0$ Where did those numbers come from?

If you plug in n= 15 you get ​a=-1/3  and the question asks for the positive integers a and n