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

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

Commented by aliyn last updated on 19/Oct/21

$$\frac{2}{7}+\frac{4}{13}+\frac{6}{19}=(\frac{2}{7}+\frac{2}{13}+\frac{2}{19})+(\frac{2}{13}+\frac{4}{19})$$$$$$$$=2a+(\frac{2}{13}+\frac{2}{19})+\left(\frac{2}{19}\right)$$$$$$$$=2a+(2a-\frac{2}{7})+(2a-\frac{2}{7}-\frac{2}{13})$$$$$$$$=6a-\frac{4}{7}-\frac{2}{13}$$$$$$$$⟨\mathit{M}.\mathit{T}⟩$$

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

$$whynot:$$$$\frac{2}{7}+\frac{4}{13}+\frac{6}{19}=\frac{1572}{1729}+a\underset{k=0}{\sum ^{100}}{(-1)}^k{C}_k^{100}$$$$or$$$$\frac{2}{7}+\frac{4}{13}+\frac{6}{19}=\frac{1572}{471}a$$$$.....$$

Answered by apriadodir last updated on 19/Oct/21

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

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