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Question Number 173776 by AgniMath last updated on 18/Jul/22

Answered by thfchristopher last updated on 18/Jul/22

$$\mathrm{When}n=1,$$$$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{1\times 2}{2}$$$$=1$$$$=\mathrm{L}.\mathrm{H}.\mathrm{S}.$$$$\mathrm{Assume}n=k\mathrm{is}\mathrm{true},$$$$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{k(k+1)}{2}$$$$\mathrm{When}n=k+1,$$$$\mathrm{R}.\mathrm{H}.\mathrm{S}.=\frac{k(k+1)}{2}+k+1$$$$=\frac{k^2+k+2k+2}{2}$$$$=\frac{k^2+3k+2}{2}$$$$=\frac{(k+1)(k+2)}{2}$$$$\therefore n=k+1\mathrm{is}\mathrm{true}.$$$$\mathrm{Inductively},n\mathrm{is}\mathrm{true}\mathrm{for}\mathrm{all}\mathrm{natural}\mathrm{values}.$$

Answered by MJS_new last updated on 18/Jul/22

$$1.n\mathrm{is}\mathrm{even}$$$$1234...$$$$nn-1n-2n-3...$$$$============$$$$n+1n+1n+1n+1...\frac{n}{2}\mathrm{times}$$$$\Rightarrow \mathrm{sum}\mathrm{is}\frac{n}{2}(n+1)$$$$2.n\mathrm{is}\mathrm{uneven}$$$$1234...$$$$nn-1n-2n-3...$$$$============$$$$n+1n+1n+1n+1...\frac{n-1}{2}\mathrm{times}\mathrm{and}\frac{n+1}{2}\mathrm{is}‘‘\mathrm{left}{\mathrm{alone}}^{″}$$$$\Rightarrow \mathrm{sum}\mathrm{is}\frac{n-1}{2}(n+1)+\frac{1}{2}(n+1)=\frac{n}{2}(n+1)$$

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