Induction Q (1 Viewer)

[Sonata]

sigmasign n(n+1) = 1/3n(n+1)(n+2)

im basically stuck at the 3rd step where u have to sub in a part from the 2nd part, i cant seem to find anything that equals it -.-

 

[Trebla]

Assume Σ(k)(k + 1) = [k(k + 1)(k + 2)]/3
Need to prove Σ(k + 1)(k + 2) = [(k + 1)(k + 2)(k + 3)]/3
LHS = Σ(k + 1)(k + 2)
= [Σ(k)(k + 1)] + (k + 1)(k + 2)
= [k(k + 1)(k + 2)]/3 + (k + 1)(k + 2) by assumption
= (k + 1)(k + 2)[k/3 + 1]
= [(k + 1)(k + 2)(k + 3)]/3
= RHS

 

[Sonata]

where did u get the part from [Σ(k)(k + 1)] ? (the 1 under LHS)

 

[Sonata]

where did u get the part from [Σ(k)(k + 1)] ? (the 1 under LHS)

 

[Trebla]

LHS = Σ(k + 1)(k + 2)
= 1 × 2 + 2 × 3 + 3 × 4 +.......+ k × (k + 1) + (k + 1)(k + 2)
= [Σ(k)(k + 1)] + (k + 1)(k + 2)

basically summarise the whole sum except the last term into a sigma:
i.e. 1 × 2 + 2 × 3 + 3 × 4 +.......+ k × (k + 1) = [Σ(k)(k + 1)]