Question

Simplify ([k+1]^2*([k+1]+1)^2)/4 to (k^2*[k+1]^2)/4. This is part of a mathematical induction problem, but I don't know how to do the algebra! Please show steps :)

Answer 1

k=0 or -1 :)

Answer 2

\[\frac{(k+1)^2 \times([k+1]+1)^2}{4} = \frac{k^2\times (k+1)^2}4\]?

Answer 3

i think ur original problem is to prove\[1^3+2^3+...+n^3=\frac{n^2(n+1)^2}{4}\]for all \(n \in \mathbb{N}\)

Answer 4

so ur question must be prove that\[\frac{(k+1)^2 \times([k+1]+1)^2}{4} = \frac{k^2\times (k+1)^2}4+(k+1)^3\]

Answer 5

Yes, that is what I am trying to solve mukushla. It's a problem from Spivak, I know how to do the induction but not the algebra. I have been trying to find an algebraic way to make them equal but no success :(

Answer 6

\[\frac{k^2\times (k+1)^2}4+(k+1)^3=\frac{k^2\times (k+1)^2}4+\frac{4(k+1)^3}{4}=\frac{(k+1)^2}{4}(k^2+4(k+1))=? \]

Answer 7

Thanks! Understand it now

Answer 8

welcome :)