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Question Number 94096 by Rio Michael last updated on 16/May/20

$$\mathrm{The}\mathrm{result}\mathrm{of}\mathrm{adding}\mathrm{the}\mathrm{odd}\mathrm{natural}\mathrm{numbers}\mathrm{is}:$$$$1=1$$$$1+3=4$$$$1+3+5=9$$$$1+3+5+7=16$$$$1+3+5+7+9=25$$$$\mathrm{show}\mathrm{that}\mathrm{from}\mathrm{this}\mathrm{result},\underset{i=1}{\sum ^n}(2i-1)=n^2.$$

Commented by Kunal12588 last updated on 16/May/20

$$sumofA.P.$$

Answered by Kunal12588 last updated on 16/May/20

$$\underset{i=1}{\sum ^n}(2i-1)=2\underset{i=1}{\sum ^n}i-n=2\times \frac{n(n+1)}{2}-n$$$$=n^2+n-n=n^2$$

Commented by Rio Michael last updated on 16/May/20

$$\mathrm{perfect}$$

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