Question

Use mathematical induction to prove the statement is true for all positive integers n, or show why it is false.

Answer 1

1st step: is the formula true for N=1 ?

when N=1 , the sum of the numbers is just 1^2 or 1
does the formula give 1 when you plug N=1 into it?

Answer 2

yes

Answer 3

that is the "base case" for induction. You showed the formula works for 1

now assume the formula works for all N ≤ K
does the formula work for K+1 ?

Answer 4

not sure

Answer 5

the last term in the series is (3n-2)^2 . We can set n= K (K and below we assume the formula works)
(3k-2)^2 is the last term in the series. If we use the formula, the series adds up to

k(6k^2-3k-1)/2

now add one more term: for (k+1)
the next term is now ( 3 (k+1) -2 )^2
and the sum of the series is k(6k^2-3k-1)/2 + ( 3 (k+1) -2 )^2

we have to show that that mess equals what the formula says the answer should be:

(k+1) ( 6 (k+1)^2 -3(k+1) -1) / 2

Answer 6

so its true? @phi

Answer 7

i got it, true

Answer 8

Well, that is the whole point, to simplify both expressions and show that they are equal.
It is a great exercise in showing off your algebra skills.