Mathematical induction

[Serenstity]

Hi,

There's a step in this question on simple mathematical induction that has really perplexed me, and I was hoping someone could help
.

I can follow it until the first line of 'Proof:'. How did they get '+(k+1)²'?
^Thanks :wink:

 

[Ishuda]

Hi,

There's a step in this question on simple mathematical induction that has really perplexed me, and I was hoping someone could help
.

I can follow it until the first line of 'Proof:'. How did they get '+(k+1)²'?
^Thanks :wink:

p(k+1) is the sum from 1 to k+1 = \(\displaystyle \overset{k+1}{\underset {r=1}{\Sigma}}\, r^2\) which is
the sum from 1 to k = \(\displaystyle \overset{k}{\underset {r=1}{\Sigma}}\, r^2\)
plus the (k+1)^st term = (k+1)²

 

[Deleted member 4993]

Hi,

There's a step in this question on simple mathematical induction that has really perplexed me, and I was hoping someone could help
.

I can follow it until the first line of 'Proof:'. How did they get '+(k+1)²'?
^Thanks :wink:

\(\displaystyle \displaystyle{\sum_{r=1}^{k+1}r^2}\)

= 1 + 2² + 3² + 4² + .... + (k-1)² + k² + (k+1)²

= \(\displaystyle \displaystyle{\sum_{r=1}^{k}r^2}\) + (k+1)²

Watch the upper limits of the summations and you should work it out with paper and pencil instead of staring at the page...