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 :)
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
$$\small{\text{$ 2\le n\le14 \text{ and } a > 0 $}}\\
\small{\text{
$ n= 2\quad a=49.500000 $ }} \\ \small{\text{
$ n= 3\quad a=32.333333 $ }} $\\$ \small{\text{
$ n= 4\quad a=23.500000 $ }} $\\$ \small{\text{
$\textcolor[rgb]{1,0,0}{n= 5\quad a=18.000000} $ }} $\\$ \small{\text{
$ n= 6\quad a=14.166667 $ }} $\\$ \small{\text{
$ n= 7\quad a=11.285714 $ }} $\\$ \small{\text{
$ \textcolor[rgb]{1,0,0}{n= 8\quad a=9.000000 }$ }} $\\$ \small{\text{
$n= 9\quad a=7.111111 $ }} $\\$ \small{\text{
$n=10\quad a=5.500000 $ }} $\\$ \small{\text{
$n=11\quad a=4.090909 $ }} $\\$ \small{\text{
$n=12\quad a=2.833333 $ }} $\\$ \small{\text{
$n=13\quad a=1.692308 $ }} $\\$ \small{\text{
$n=14\quad a=0.642857 $ }} $\\$ \small{\text{
The only 2 solutions for $(a,n)$ are $ (18,5),\ (9,8)$
}} $\\$
\small{\text{
$
\textcolor[rgb]{1,0,0}{18}+19+20+21+22 = 100 \quad $ and $\quad \textcolor[rgb]{1,0,0}{9}+10+11+12+13+14+15+16 = 100
$
}}$$
Thanks Heureka
My answer is wrong - the error was right near the beginning.
I am sure that Heureka's answer is perfect.