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## Homework Statement

Summation of i(i + 1) (with i going from i = 2 to i = n-1) = n(n-1)(n=1) / 3

a. Write P(2). Is P(2) true?

b. Write P(k)

c. Write P(k+1)

d. Prove by mathematical induction that the formula holds true for all integers

n [tex]\geq[/tex] 2

## Homework Equations

N/A

## The Attempt at a Solution

a. P(2): i(i+ 1) + ... + (n-1)[(n-1)+1] = n(n-1)(n+1) / 3

= (2-1)[(2-1) + 1] = 2(2-1)(2 + 1) / 3

P(2): 2 = 2

P(2) is true

b. P(k): ...+ (k-1)[(k-1)+1] = k(k-1)(k+1) / 3

P(k) = k(k-1) = k(k

^{2}- 1)/3

c. P(k+1): (k+1)(k-1) = (k+1)[(k+1)

^{2}- 1) / 3

= k

^{2}- 1 = k

^{3}+ 3k

^{2}+ 2k / 3

d. Left-Hand side of P(k+1) = i(i + 1) + ... + (k+1)(k-1)

= i(i + 1) + ... + k(k-1) + (k+1)(k-1)

= k(k

^{2}- 1)/3 + 3k

^{2}-3 / 3

= 4k

^{2}- 4

Right-Hand side of P(k+1) = k

^{3}+ 3k

^{2}+ 2k / 3

4k

^{2}- 4 = k

^{3}+ 3k

^{2}+ 2k / 3

I've went over this several times and it doesn't work out so I am obviously doing something

wrong, but I am not sure where I am making the mistake.