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Question Number 157057 by amin96 last updated on 19/Oct/21

(1/7)+(1/(13))+(1/(19))=a find (2/7)+(4/(13))+(6/(19))=?

$$\frac{\mathrm{1}}{\mathrm{7}}+\frac{\mathrm{1}}{\mathrm{13}}+\frac{\mathrm{1}}{\mathrm{19}}={a} \\ $$$${find}\:\frac{\mathrm{2}}{\mathrm{7}}+\frac{\mathrm{4}}{\mathrm{13}}+\frac{\mathrm{6}}{\mathrm{19}}=? \\ $$

Commented by aliyn last updated on 19/Oct/21

(2/7)+(4/(13))+(6/(19))=((2/7) + (2/(13)) +(2/(19)) )+((2/(13))+(4/(19))) = 2 a + ( (2/(13)) + (2/(19)) ) + ( (2/(19))) = 2 a + (2a − (2/7)) + (2a−(2/7)−(2/(13))) = 6a −(4/7) − (2/(13)) ⟨ M . T ⟩

$$\frac{\mathrm{2}}{\mathrm{7}}+\frac{\mathrm{4}}{\mathrm{13}}+\frac{\mathrm{6}}{\mathrm{19}}=\left(\frac{\mathrm{2}}{\mathrm{7}}\:+\:\frac{\mathrm{2}}{\mathrm{13}}\:+\frac{\mathrm{2}}{\mathrm{19}}\:\right)+\left(\frac{\mathrm{2}}{\mathrm{13}}+\frac{\mathrm{4}}{\mathrm{19}}\right) \\ $$$$ \\ $$$$=\:\mathrm{2}\:{a}\:+\:\left(\:\frac{\mathrm{2}}{\mathrm{13}}\:+\:\frac{\mathrm{2}}{\mathrm{19}}\:\right)\:+\:\left(\:\frac{\mathrm{2}}{\mathrm{19}}\right) \\ $$$$ \\ $$$$=\:\mathrm{2}\:{a}\:+\:\left(\mathrm{2}{a}\:−\:\frac{\mathrm{2}}{\mathrm{7}}\right)\:+\:\left(\mathrm{2}{a}−\frac{\mathrm{2}}{\mathrm{7}}−\frac{\mathrm{2}}{\mathrm{13}}\right) \\ $$$$ \\ $$$$=\:\mathrm{6}{a}\:−\frac{\mathrm{4}}{\mathrm{7}}\:−\:\frac{\mathrm{2}}{\mathrm{13}} \\ $$$$ \\ $$$$\langle\:\boldsymbol{{M}}\:.\:\boldsymbol{{T}}\:\:\rangle \\ $$

Commented by mr W last updated on 19/Oct/21

why not: (2/7)+(4/(13))+(6/(19))=((1572)/(1729))+aΣ_(k=0) ^(100) (−1)^k C_k ^(100) or (2/7)+(4/(13))+(6/(19))=((1572)/(471))a .....

$${why}\:{not}: \\ $$$$\frac{\mathrm{2}}{\mathrm{7}}+\frac{\mathrm{4}}{\mathrm{13}}+\frac{\mathrm{6}}{\mathrm{19}}=\frac{\mathrm{1572}}{\mathrm{1729}}+{a}\underset{{k}=\mathrm{0}} {\overset{\mathrm{100}} {\sum}}\left(−\mathrm{1}\right)^{{k}} {C}_{{k}} ^{\mathrm{100}} \\ $$$${or} \\ $$$$\frac{\mathrm{2}}{\mathrm{7}}+\frac{\mathrm{4}}{\mathrm{13}}+\frac{\mathrm{6}}{\mathrm{19}}=\frac{\mathrm{1572}}{\mathrm{471}}{a} \\ $$$$..... \\ $$

Answered by apriadodir last updated on 19/Oct/21

answers: (2/7)+(4/(13))+(6/(19))=((2/7)+(2/(13))+(2/(19)))+((2/(13))+(2/(19)))+(2/(19)) = 2a +2a−(2/7)+2a−(2/7)−(2/(13)) = 6a−(4/7)−(2/(13))

$$\mathrm{answers}: \\ $$$$\frac{\mathrm{2}}{\mathrm{7}}+\frac{\mathrm{4}}{\mathrm{13}}+\frac{\mathrm{6}}{\mathrm{19}}=\left(\frac{\mathrm{2}}{\mathrm{7}}+\frac{\mathrm{2}}{\mathrm{13}}+\frac{\mathrm{2}}{\mathrm{19}}\right)+\left(\frac{\mathrm{2}}{\mathrm{13}}+\frac{\mathrm{2}}{\mathrm{19}}\right)+\frac{\mathrm{2}}{\mathrm{19}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{2a}\:+\mathrm{2a}−\frac{\mathrm{2}}{\mathrm{7}}+\mathrm{2a}−\frac{\mathrm{2}}{\mathrm{7}}−\frac{\mathrm{2}}{\mathrm{13}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\:\mathrm{6a}−\frac{\mathrm{4}}{\mathrm{7}}−\frac{\mathrm{2}}{\mathrm{13}} \\ $$

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