Series exercise based on raising the exponent to the power of 2!

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wolly

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How does [math]\left[(a+\frac{1}{n})^{2}+(a+\frac{2}{n})^{2}+...+(a+\frac{n}{n})^{2}\right][/math] = [math](n-1)a^{2}+2*a\left(\frac{1}{n}+\frac{2}{n}+...+\frac{n-1}{n}\right)+\frac{1^{2}}{n^{2}}+\frac{2^2}{n^{2}}+...+\frac{(n-1)^{2}}{n^{2}}[/math]?

I tried something but I don't know if this is correct and it's not the same with the upper image!


[math]\frac{(an+1)^{2}}{n^{2}}+\frac{(an+2)^{2}}{n^{2}}+\frac{(an+3)^{2}}{n^{2}}+...+\frac{(an+n-1)^{2}}{n^{2}}+\frac{(an+n)^{2}}{n^{2}}[/math] = [math]\frac{1}{n^{2}}\left((a^{2}n^{2}+2an+1)+(a^{2}n^{2}+4an+4)+(a^{2}n^{2}+6an+9)+...+(a^{2}n^{2}+2an(n-1)+(n-1)^{2}+(a^{2}n^{2}+2an*n+n^{2})\right)[/math]
[math]\frac{(an+1)^{2}}{n^{2}}+\frac{(an+2)^{2}}{n^{2}}+\frac{(an+3)^{2}}{n^{2}}+...+\frac{(an+n-1)^{2}}{n^{2}}+\frac{(an+n)^{2}}{n^{2}}[/math]

[math]\frac{1}{n^{2}}\left((a^{2}n^{2}+2an+1)+(a^{2}n^{2}+4an+4)+(a^{2}n^{2}+6an+9)+...+((a^{2}n^{2}+2an(n-1)+(n-1)^{2})+(a^{2}n^{2}+2an*n+n^{2})\right)[/math]
Is it correct what I wrote??
 
wolly said:
How does [math]\left[(a+\frac{1}{n})^{2}+(a+\frac{2}{n})^{2}+...+(a+\frac{n}{n})^{2}\right][/math] = [math](n-1)a^{2}+2*a\left(\frac{1}{n}+\frac{2}{n}+...+\frac{n-1}{n}\right)+\frac{1^{2}}{n^{2}}+\frac{2^2}{n^{2}}+...+\frac{(n-1)^{2}}{n^{2}}[/math]?

I tried something but I don't know if this is correct and it's not the same with the upper image!


[math]\frac{(an+1)^{2}}{n^{2}}+\frac{(an+2)^{2}}{n^{2}}+\frac{(an+3)^{2}}{n^{2}}+...+\frac{(an+n-1)^{2}}{n^{2}}+\frac{(an+n)^{2}}{n^{2}}[/math] = [math]\frac{1}{n^{2}}\left((a^{2}n^{2}+2an+1)+(a^{2}n^{2}+4an+4)+(a^{2}n^{2}+6an+9)+...+(a^{2}n^{2}+2an(n-1)+(n-1)^{2}+(a^{2}n^{2}+2an*n+n^{2})\right)[/math]
[math]\frac{(an+1)^{2}}{n^{2}}+\frac{(an+2)^{2}}{n^{2}}+\frac{(an+3)^{2}}{n^{2}}+...+\frac{(an+n-1)^{2}}{n^{2}}+\frac{(an+n)^{2}}{n^{2}}[/math]

[math]\frac{1}{n^{2}}\left((a^{2}n^{2}+2an+1)+(a^{2}n^{2}+4an+4)+(a^{2}n^{2}+6an+9)+...+((a^{2}n^{2}+2an(n-1)+(n-1)^{2})+(a^{2}n^{2}+2an*n+n^{2})\right)[/math]
Is it correct what I wrote??
Click to expand...

It looks like the right hand side in the first equality is missing a couple of items (typeset in boldface below):
[math](n-1)a^{2}+2*a\left(\frac{1}{n}+\frac{2}{n}+...+\frac{n-1}{n} +\mathbf{\frac{n}{n}} \right) +\frac{1^{2}}{n^{2}}+\frac{2^2}{n^{2}}+...+\frac{(n-1)^{2}}{n^{2}} \mathbf{+\frac{n^2}{n^2}} [/math]
 
blamocur said:
It looks like the right hand side in the first equality is missing a couple of items (typeset in boldface below):
[math](n-1)a^{2}+2*a\left(\frac{1}{n}+\frac{2}{n}+...+\frac{n-1}{n} +\mathbf{\frac{n}{n}} \right) +\frac{1^{2}}{n^{2}}+\frac{2^2}{n^{2}}+...+\frac{(n-1)^{2}}{n^{2}} \mathbf{+\frac{n^2}{n^2}} [/math]?
Click to expand...
And I missed another one: shouldn't it be [imath]na^2[/imath] instead of [imath](n-1)a^2[/imath] in the same expression?
 
wolly said:
Is it correct what I wrote??
Click to expand...
It looks correct to me, but you want to group together multiples of [imath]n^2[/imath], multiples of [imath]n[/imath] and "free" elements, i.e., not containing [imath]n[/imath]'s.
 
[math]\frac{1}{n^{2}}*\left(n^{2}*a^{2}+n^{2}*a^{2}+....+n^{2}a^{2}\right)[/math]= [math]a^{2}+a^{2}+...+a^{2}[/math] and from here?????
 
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