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Question Number 53515 by 0955083339 last updated on 22/Jan/19

What is the sum of ten odd numbers  and eleven even numbers?

$$\mathrm{What}\:\mathrm{is}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{ten}\:\mathrm{odd}\:\mathrm{numbers} \\ $$$$\mathrm{and}\:\mathrm{eleven}\:\mathrm{even}\:\mathrm{numbers}? \\ $$

Answered by tanmay.chaudhury50@gmail.com last updated on 23/Jan/19

S_(odd) =1+3+5+...+[1+(10−1)2]  S_(odd) =1+3+5+...+19  S_(odd) =19+17+15+...+1  add  2S_(odd) =20+20+20...ten times  2S_(odd) =20×10  S_(odd) =100  or direct formula   S=(n/2)[2a+(n−1)d]  =((10)/2)[2×1+(10−1)2]  =100  S_(even) =2+4+6+...+[2+(11−1)2]  S_(even) =2+4+6+...+22  S_(even) =22+20+18+...+2  add  2S_(even) =24+24+24...eleven times  2S_(even) =24×11  S_(even) =((24×11)/2)=132  or direct formula  S_(even) =((11)/2)[2×2+(11−1)2]  S_(even) =((11)/2)[24]=132

$${S}_{{odd}} =\mathrm{1}+\mathrm{3}+\mathrm{5}+...+\left[\mathrm{1}+\left(\mathrm{10}−\mathrm{1}\right)\mathrm{2}\right] \\ $$$${S}_{{odd}} =\mathrm{1}+\mathrm{3}+\mathrm{5}+...+\mathrm{19} \\ $$$${S}_{{odd}} =\mathrm{19}+\mathrm{17}+\mathrm{15}+...+\mathrm{1} \\ $$$${add} \\ $$$$\mathrm{2}{S}_{{odd}} =\mathrm{20}+\mathrm{20}+\mathrm{20}...{ten}\:{times} \\ $$$$\mathrm{2}{S}_{{odd}} =\mathrm{20}×\mathrm{10} \\ $$$${S}_{{odd}} =\mathrm{100} \\ $$$${or}\:{direct}\:{formula}\: \\ $$$${S}=\frac{{n}}{\mathrm{2}}\left[\mathrm{2}{a}+\left({n}−\mathrm{1}\right){d}\right] \\ $$$$=\frac{\mathrm{10}}{\mathrm{2}}\left[\mathrm{2}×\mathrm{1}+\left(\mathrm{10}−\mathrm{1}\right)\mathrm{2}\right] \\ $$$$=\mathrm{100} \\ $$$${S}_{{even}} =\mathrm{2}+\mathrm{4}+\mathrm{6}+...+\left[\mathrm{2}+\left(\mathrm{11}−\mathrm{1}\right)\mathrm{2}\right] \\ $$$${S}_{{even}} =\mathrm{2}+\mathrm{4}+\mathrm{6}+...+\mathrm{22} \\ $$$${S}_{{even}} =\mathrm{22}+\mathrm{20}+\mathrm{18}+...+\mathrm{2} \\ $$$${add} \\ $$$$\mathrm{2}{S}_{{even}} =\mathrm{24}+\mathrm{24}+\mathrm{24}...{eleven}\:{times} \\ $$$$\mathrm{2}{S}_{{even}} =\mathrm{24}×\mathrm{11} \\ $$$${S}_{{even}} =\frac{\mathrm{24}×\mathrm{11}}{\mathrm{2}}=\mathrm{132} \\ $$$${or}\:{direct}\:{formula} \\ $$$${S}_{{even}} =\frac{\mathrm{11}}{\mathrm{2}}\left[\mathrm{2}×\mathrm{2}+\left(\mathrm{11}−\mathrm{1}\right)\mathrm{2}\right] \\ $$$${S}_{{even}} =\frac{\mathrm{11}}{\mathrm{2}}\left[\mathrm{24}\right]=\mathrm{132} \\ $$

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