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Question Number 13121 by tawa tawa last updated on 14/May/17

Find the value of :  (2/(15)) + (2/(35)) + (2/(63)) + (2/(99)) + ... + (2/(9999))

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\::\:\:\frac{\mathrm{2}}{\mathrm{15}}\:+\:\frac{\mathrm{2}}{\mathrm{35}}\:+\:\frac{\mathrm{2}}{\mathrm{63}}\:+\:\frac{\mathrm{2}}{\mathrm{99}}\:+\:...\:+\:\frac{\mathrm{2}}{\mathrm{9999}} \\ $$

Answered by nume1114 last updated on 15/May/17

    (2/(15))+(2/(35))+(2/(63))+(2/(99))+...+(2/(9999))  =(2/(3×5))+(2/(5×7))+(2/(7×9))+...+(2/(99×101))  =Σ_(k=1) ^(49) (2/((2k+1)(2k+3)))  =Σ_(k=1) ^(49) ((1/(2k+1))−(1/(2k+3)))  =((1/3)−(1/5))+((1/5)−(1/7))+((1/7)−(1/9))+...+((1/(99))−(1/(101)))  =(1/3)−(1/(101))  =((98)/(303))

$$\:\:\:\:\frac{\mathrm{2}}{\mathrm{15}}+\frac{\mathrm{2}}{\mathrm{35}}+\frac{\mathrm{2}}{\mathrm{63}}+\frac{\mathrm{2}}{\mathrm{99}}+...+\frac{\mathrm{2}}{\mathrm{9999}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}×\mathrm{5}}+\frac{\mathrm{2}}{\mathrm{5}×\mathrm{7}}+\frac{\mathrm{2}}{\mathrm{7}×\mathrm{9}}+...+\frac{\mathrm{2}}{\mathrm{99}×\mathrm{101}} \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{\mathrm{49}} {\sum}}\frac{\mathrm{2}}{\left(\mathrm{2}{k}+\mathrm{1}\right)\left(\mathrm{2}{k}+\mathrm{3}\right)} \\ $$$$=\underset{{k}=\mathrm{1}} {\overset{\mathrm{49}} {\sum}}\left(\frac{\mathrm{1}}{\mathrm{2}{k}+\mathrm{1}}−\frac{\mathrm{1}}{\mathrm{2}{k}+\mathrm{3}}\right) \\ $$$$=\left(\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{5}}\right)+\left(\frac{\mathrm{1}}{\mathrm{5}}−\frac{\mathrm{1}}{\mathrm{7}}\right)+\left(\frac{\mathrm{1}}{\mathrm{7}}−\frac{\mathrm{1}}{\mathrm{9}}\right)+...+\left(\frac{\mathrm{1}}{\mathrm{99}}−\frac{\mathrm{1}}{\mathrm{101}}\right) \\ $$$$=\frac{\mathrm{1}}{\mathrm{3}}−\frac{\mathrm{1}}{\mathrm{101}} \\ $$$$=\frac{\mathrm{98}}{\mathrm{303}} \\ $$

Commented by tawa tawa last updated on 15/May/17

God bless you sir.

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir}. \\ $$

Commented by nume1114 last updated on 15/May/17

oh,you are right sir.I′ve corrected it.

$${oh},{you}\:{are}\:{right}\:{sir}.{I}'{ve}\:{corrected}\:{it}. \\ $$

Commented by RasheedSindhi last updated on 15/May/17

NiCE!

$$\mathcal{N}{i}\mathcal{CE}! \\ $$

Commented by tawa tawa last updated on 15/May/17

but the answer = ((98)/(303))  mistake in your L.C.M

$$\mathrm{but}\:\mathrm{the}\:\mathrm{answer}\:=\:\frac{\mathrm{98}}{\mathrm{303}} \\ $$$$\mathrm{mistake}\:\mathrm{in}\:\mathrm{your}\:\mathrm{L}.\mathrm{C}.\mathrm{M} \\ $$

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