Question

: prove by mathematical induction

∑ 𝑖(𝑖 + 1)
𝑛−1
𝑖=0 =
𝑛(𝑛−1)(𝑛+1)
3
for all integers n ≥ 2

Answer 1

geez - learn to type fractions.
case i=0
0*1 = 0
0(-1)(1)/3 = 0
so, P(0) is true

so, assume true for i=0..n-1
The next term is (with i=n)
n(n+1)
add that to both sides and you now just need to show that
∑(0)(1) ... (n-1)(n) + n(n+1) = (n-1)(n)(n+1)/3 + n(n+1)

(n-1)(n)(n+1)/3 + n(n+1) = ((n-1)(n)(n+1) + 3n(n+1))/3
= (n^3 - n + 3n^2 + 3n)/3
= (n^3 + 3n^2 + 2n)/3
= n(n^2+3n+2)/3
= n(n+1)(n+2)/3
= ((n+1)-1)(n+1)((n+1)+1)/3
= P(n)
so, P(n-1) ==> P(n)