Summing Series (Sigma Notation)

[I-need-help]

Homework Statement

a) ⁿƩr²(r-1)_r=1

Homework Equations

Using the summation series formulae...

The Attempt at a Solution

So far I have got:

r²(r-1) = r³-r²

Ʃr³ = $\frac{1}{4}$n² (n+1)²

Ʃr² = $\frac{1}{6}$n(n+1)(2n+1)

Therefore,

Ʃr³-r² = $\frac{1}{4}$n² (n+1)² - $\frac{1}{6}$n(n+1)(2n+1)

But how do I subtract these two massive things from each other??

Thanks.

 

[eumyang]

They aren't that massive. Rewrite them as fractions:
$$\frac{n^2(n+1)^2}{4}-\frac{n(n+1)(2n+1)}{6}$$
Find a common denominator, rewrite both fractions so that both have that common denominator, and then subtract. It will be possible to factor things out.

 

[HallsofIvy]

It's pretty much just subtracting fractions. Get a common denominator:
The least common denominator of 4 and 6 is 12. To change the first denominator to 12, multiply both numerator and denominator by 3.
$$\frac{n^2(n+1)^2}{4}= \frac{3n^2(n+1)^2}{12}$$

To change the second denominator to 12, multiply both numerator and denominator by 4.
$$\frac{4n(n+1)(2n+1)}{12}$$

To subtract now, subtract the numerators:
$$\frac{3n^2(n+1)^2}{12}- \frac{4n(n+1)(2n+1)}{12}$$

You will need to multiply those:
$3n^2(n+1)^2= 3n^2(n^2+ 2n+ 1)= 3n^4+ 6n^3+ 3n^2$
$4n(n+1)(2n+1)= 4n(2n^2+ 3n+ 1)= 8n^3+ 12n^2+ 4n$

and now subtract.

 

[I-need-help]

They aren't that massive. Rewrite them as fractions:
$$\frac{n^2(n+1)^2}{4}-\frac{n(n+1)(2n+1)}{6}$$
Find a common denominator, rewrite both fractions so that both have that common denominator, and then subtract. It will be possible to factor things out.

Yeah ok, I've done that, but the textbook answers want it to be "factorised", and I'm unsure how to do that after subtracting... Thanks for your help.

 

[I-need-help]

@HallsOfIvy

Hmm thanks, for the help, I'll try to factorise those now, since the book wants it to be factorised.

 

[HallsofIvy]

You will have a quadratic factor that cannot be factored further (with integer coefficients).