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Given-1-3-5-7-16-we-know-that-16-4-2-and-4-is-the-half-of-8-which-is-the-successor-of-7-conjecture-the-result-of-this-sum-1-3-5-7-25-




Question Number 116683 by mathocean1 last updated on 05/Oct/20
Given 1+3+5+7=16   we know that  16=4^2   and 4 is the half of 8 which is the   successor of 7.    conjecture the result of this sum:  1+3+5+7+...+25
$$\mathrm{Given}\:\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}=\mathrm{16}\:\:\:\mathrm{we}\:\mathrm{know}\:\mathrm{that} \\ $$$$\mathrm{16}=\mathrm{4}^{\mathrm{2}} \:\:\mathrm{and}\:\mathrm{4}\:\mathrm{is}\:\mathrm{the}\:\mathrm{half}\:\mathrm{of}\:\mathrm{8}\:\mathrm{which}\:\mathrm{is}\:\mathrm{the}\: \\ $$$$\mathrm{successor}\:\mathrm{of}\:\mathrm{7}. \\ $$$$ \\ $$$$\mathrm{conjecture}\:\mathrm{the}\:\mathrm{result}\:\mathrm{of}\:\mathrm{this}\:\mathrm{sum}: \\ $$$$\mathrm{1}+\mathrm{3}+\mathrm{5}+\mathrm{7}+…+\mathrm{25} \\ $$
Answered by Dwaipayan Shikari last updated on 05/Oct/20
13 terms ,and 13^2 =169    Generally  T_n =(2n−1)  Σ_(n=1) ^n (2n−1)=n(n+1)−n=n^2
$$\mathrm{13}\:{terms}\:,{and}\:\mathrm{13}^{\mathrm{2}} =\mathrm{169} \\ $$$$ \\ $$$${Generally}\:\:{T}_{{n}} =\left(\mathrm{2}{n}−\mathrm{1}\right) \\ $$$$\underset{{n}=\mathrm{1}} {\overset{{n}} {\sum}}\left(\mathrm{2}{n}−\mathrm{1}\right)={n}\left({n}+\mathrm{1}\right)−{n}={n}^{\mathrm{2}} \\ $$
Commented by mathocean1 last updated on 05/Oct/20
thank you sir
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{sir} \\ $$$$ \\ $$
Answered by $@y@m last updated on 06/Oct/20
Successor of 25 is 26.  Half of 26 is 13.  ∴ The given sum=13^2 =169
$${Successor}\:{of}\:\mathrm{25}\:{is}\:\mathrm{26}. \\ $$$${Half}\:{of}\:\mathrm{26}\:{is}\:\mathrm{13}. \\ $$$$\therefore\:{The}\:{given}\:{sum}=\mathrm{13}^{\mathrm{2}} =\mathrm{169} \\ $$

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