Quandaries
and Queries
Use induction to show that
1 2 + 2 2 + .....+n 2
= (n 3)/3 + (n 2)/2 + n/6
Thanks Pooh
Hi,
Getting started: n=1
R.S. = (1 3)/3 + (1 2)/2 + 1/6 = 1/3 + 1/2 + 1/6 = 2/6 + 3/6 + 1/6 = 6/6 = 1
Keeping induction going: assume true for n=k and prove
true for n=k+1 True for n=k means
1 2 + 2 2 + .....+k 2
= (k 3)/3 + (k 2)/2 + k/6
Now consider n=k+1
= (k 3)/3 + (k 2)/2 + k/6 + (k+1) 2
= 2k 3/6 + 3k 2/6 + k/6 + (6k 2 + 12k + 6)/6
= (2k 3 + 9k 2 + 13k + 6)/6
R.S.
= (k 3 + 3k 2 + 3k + 1)/3 + (k 2 + 2k + 1)/2 + (k+1)/6
= (2k 3 + 6k 2 + 6k + 2)/6 + (3k 2 + 6k + 3)/6 + (k+1)/6
= (2k 3 + 9k 2 + 13k + 6)/6