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

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Part (a): Find the sumin terms of  and

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

Guest Jan 20, 2015

#3
+20593
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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

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

heureka  Jan 20, 2015
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http://web2.0calc.com/questions/instructions-on-reposting_1

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

Melody  Jan 20, 2015
#3
+20593
+5

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

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

heureka  Jan 20, 2015
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Thanks Heureka

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

I am sure that Heureka's answer is perfect.

Melody  Jan 20, 2015
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Heureka, could you please explain how you got $$2\leq n\leq14$$ and $$a>0$$ Where did those numbers come from?

Guest Jan 22, 2016
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If you plug in n= 15 you get ​a=-1/3  and the question asks for the positive integers a and n

Guest Jan 31, 2017