<s>[INST] Prove $(1+2+...+k)^2 = 1^3 + ... + k^3$ using induction I need to prove that
$$(1+2+{...}+k)^2 = 1^3 + {...} + k^3$$
using induction.
So the base case holds for $0$ because $0 = 0$ (and also for $1$: $1^2 = 1^3 = 1$)
I can't prove it for $k+1$ no matter what I try! Can you give me a hint?
 [/INST] Induction on k.
Base Case: 1=1
Induction Hypothesis: $(1+2+{...}+k)^2$=$1^3 + {...} + k^3$
Based on the hypothesis: 
$(1+2+{...}+k+(k+1))^2$
=$(1+2+{...}+k)^2$+2$(1+2+{...}+k)(k+1)$+$(k+1)^2$
=$1^3 + {...} + k^3$+$k(k+1)^2$+$(k+1)^2$
=$1^3 + {...} + k^3 + (k+1)^3$
Done.
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