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Question Number 6350 by sanusihammed last updated on 24/Jun/16

$$Provebymatimaticalinduction$$$$1+8+27+......+n^3={\left[\frac{n(n+1)}{2}\right]}^2$$

Answered by prakash jain last updated on 24/Jun/16

$$n=1$$$$1={\left(\frac{1\cdot 2}{2}\right)}^2\mathrm{so}\mathrm{statement}\mathrm{is}\mathrm{true}\mathrm{for}n=1$$$$assumeitistrueforn=k$$$$\mathrm{then}\underset{i=1}{\sum ^k}{i}^3={\left[\frac{k(k+1)}{2}\right]}^2$$$$\underset{i=1}{\sum ^{k+1}}{i}^3={\left[\frac{k(k+1)}{2}\right]}^2+{(k+1)}^3={(k+1)}^2[\frac{k^2}{2^2}+k+1]$$$$=\frac{{(k+1)}^2}{2^2}[k^2+4k+k)=\frac{{(k+1)}^2{(k+2)}^2}{2^2}$$$$ifstatementistrueforn=kthenitistruefor$$$$n=k+1.$$$$◼$$

Commented by sanusihammed last updated on 25/Jun/16

$$Thankssomuch$$

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