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Part (a): Find the suma + (a + 1) + (a + 2) + \dots + (a + n - 1)in terms of a and n. 

Part (b): Find all pairs of positive integers (a,n) such that n \ge 2 anda + (a + 1) + (a + 2) + \dots + (a + n - 1) = 100.

Guest Jan 20, 2015

Best Answer 

 #3
avatar+18843 
+5

Part (a): Find the sum  s =  a + (a + 1) + (a + 2) + \dots + (a + n - 1) in terms of a and n. 

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

 

Part (b): Find all pairs of positive integers (a,n) such that n \ge 2 anda + (a + 1) + (a + 2) + \dots + (a + n - 1) = 100. 

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

heureka  Jan 20, 2015
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5+0 Answers

 #2
avatar+91510 
+5

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
avatar+18843 
+5
Best Answer

Part (a): Find the sum  s =  a + (a + 1) + (a + 2) + \dots + (a + n - 1) in terms of a and n. 

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

 

Part (b): Find all pairs of positive integers (a,n) such that n \ge 2 anda + (a + 1) + (a + 2) + \dots + (a + n - 1) = 100. 

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

heureka  Jan 20, 2015
 #4
avatar+91510 
+3

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
 #4
avatar
0

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
 #5
avatar
0

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

Guest Jan 31, 2017

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