Prove that 13+23+33+⋯+n31^3 + 2^3 + 3^3 + \cdots + n^313+23+33+⋯+n3 is | Discrete Structure 2081

Discrete Structure20815 marksMathematical induction

Prove that 13+23+33+⋯+n31^3 + 2^3 + 3^3 + \cdots + n^313+23+33+⋯+n3 is a perfect square using mathematical induction. [5]

$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$ This is a perfect square for all positive integers $n$ . --- LHS: $1^3 = 1$ RHS: $\left[\frac{1(1+1)}{2}\right]^2 = \left[\frac{1 \cdot 2}{2}\right]^2 = [1]^2 = 1$ Since LHS $=$ RHS $= 1$ , the base case holds. $\checkmark$ --- As...

$$1^3 + 2^3 + 3^3 + \cdots + n^3 = \left[\frac{n(n+1)}{2}\right]^2$$ This is a perfect square for all positive integers $n$. --- LHS: $1^3 = 1$ RHS: $\left[\frac{1(1+1)}{2}\right]^2 = \left[\frac{1 \cdot 2}{2}\right]^2 = [1]^2 = 1$ Since LHS $=$ RHS $= 1$, the base case holds. $\checkmark$ --- As...

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