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Summation Proof By Induction watch

1. (Idk how to do summation notation with LaTex)
Prove that the some of integers 1 to n in the function
__________________
So what I've done is show that it is true for n = 1, then I assumed that it is true for n = k and now I look at n = k + 1.
So: The sum of all integers in the function from 1 to k + 1 =

So that's all the easy part and I know I need to now get it to look like

But I am struggling to connect the dots.
2. (Original post by Retsek)
(Idk how to do summation notation with LaTex)
Prove that the some of integers 1 to n in the function
__________________
So what I've done is show that it is true for n = 1, then I assumed that it is true for n = k and now I look at n = k + 1.
So: The sum of all integers in the function from 1 to k + 1 =

So that's all the easy part and I know I need to now get it to look like

But I am struggling to connect the dots.
You have two terms with common factors (k+1)(k+2) so you need to "take them outside the brackets" to give

Can you continue? Factorising like this is a very common thing you need to do in induction questions.

Also, \displaystyle \sum_{r=1}^{n} in LaTeX produces

3. (Original post by Notnek)
You have two terms with common factors (k+1)(k+2) so you need to "take them outside the brackets" to give

Can you continue? Factorising like this is a very common thing you need to do in induction questions.

Also, \displaystyle \sum_{r=1}^{n} in LaTeX produces

So I managed to get minutes after making this post but I didn't realise for a solid ten minutes that I could just move the 1/3 to the k+1

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