Question

Please help me out!
(show all work)

Answer 1

first verify it holds for n=1
then you are allowed to assume it holds for n and you can start with the induction step

for n+1 instead of inserting it into the equation, add it to both sides and transform it to the original formula to proof its equality


`1^(2)+2^(2)+3^(2)+...+n^(2)+(n+1)^(2)=(n(n+1)(2n+1))/(6) +(n+1)^(2) \\ =(n(n+1)(2n+1)+6(n+1)^(2) )/(6)\\ =(n(n+1)(2n+1)+6(n+1)(n+1))/(6)\\ =((n+1)(n(2n+1)+6(n+1)))/(6)\\ =((n+1)((2n^(2)+n)+6(n+1)))/(6)\\ =((n+1)(2n^(2)+7n+6))/(6)\\ =((n+1)(n+2)(2n+3))/(6)\\ =((n+1)((n+1)+1)(2(n+1)+1))/(6)`

this equals the original formula if you had inserted n+1, proving the correctness