The-result-of-adding-the-odd-natural-numbers-is-1-1-1-3-4-1-3-5-9-1-3-5-7-16-1-3-5-7-9-25-show-that-from-this-result-i-1-n-2i-1-n-2-

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

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

Commented by Kunal12588 last updated on 16/May/20

$${sum}\:{of}\:{A}.{P}. \\ $$

Answered by Kunal12588 last updated on 16/May/20

$$\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{2}{i}−\mathrm{1}\right)=\mathrm{2}\underset{{i}=\mathrm{1}} {\overset{{n}} {\sum}}{i}−{n}=\mathrm{2}×\frac{{n}\left({n}+\mathrm{1}\right)}{\mathrm{2}}−{n} \\ $$$$={n}^{\mathrm{2}} +{n}−{n}={n}^{\mathrm{2}} \\ $$

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

$$\mathrm{perfect} \\ $$

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