Using the principle of mathematical induction prove that 12-22-n2-n3-3 for all values of n - N- Or Evaluate -16-30 i

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Proof by mathematical induction

Using the pri...

Question

Using the principle of mathematical induction prove that $(1^{2} + 2^{2} + \dots + n^{2}) > \frac{n^{3}}{3}$ for all values of n $ϵ$ N.

Or

Evaluate $\sqrt{16 - 30 i}$

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1^{2} + 2^{2} + 3^{2} + . . . . + n^{2} > \frac{n^{2}}{3}

n \in N

n \geq 1.

1^{2} + 2^{2} + 3^{2} + \dots + n^{2} > \frac{n^{3}}{3} .

Prove the following by using the principle of mathematical induction for all n ∈ N:

n \in N

1^{3} + 2^{3} + 3^{3} + \dots + n^{3} = {(\frac{n (n + 1)}{2})}^{2}

n \in N : (1 + \frac{3}{1}) (1 + \frac{5}{4}) (1 + \frac{7}{9}) . . . . . . (1 + \frac{(2 n + 1)}{n^{2}}) = {(n + 1)}^{2}

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