Question Number 56416 by gunawan last updated on 16/Mar/19 | ||
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$$\underset{{n}−\mathrm{digits}} {\left(\mathrm{666}\:....\:\mathrm{6}\right)^{\mathrm{2}} }\:+\:\underset{{n}−\mathrm{digits}} {\left(\mathrm{888}\:....\mathrm{8}\right)}\:\mathrm{is}\:\mathrm{equal}\:\mathrm{to} \\ $$ | ||
Commented by mr W last updated on 16/Mar/19 | ||
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$$=\underset{\mathrm{2}{n}\:{digits}} {\left(\mathrm{444}...\mathrm{4}\right)} \\ $$ | ||
Answered by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19 | ||
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$$\left(\mathrm{666}....\mathrm{6}\right)\leftarrow{n}\:{digit} \\ $$$$\mathrm{6}\left(\mathrm{111}...\mathrm{1}\right) \\ $$$$\mathrm{6}\left\{\mathrm{10}^{{n}−\mathrm{1}} +\mathrm{10}^{{n}−\mathrm{2}} +\mathrm{10}^{{n}−\mathrm{3}} +...+\mathrm{1}\right\} \\ $$$$\mathrm{6}×\frac{\mathrm{10}^{{n}−\mathrm{1}} \left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{10}^{{n}} }\right)}{\left(\mathrm{1}−\frac{\mathrm{1}}{\mathrm{10}}\right)}\rightarrow\frac{\mathrm{6}}{\mathrm{9}}×\mathrm{10}^{{n}} \left(\frac{\mathrm{10}^{{n}} −\mathrm{1}}{\mathrm{10}^{{n}} }\right)\rightarrow\frac{\mathrm{6}}{\mathrm{9}}\left(\mathrm{10}^{{n}} −\mathrm{1}\right) \\ $$$${so}\:{answer}\:{is} \\ $$$$\frac{\mathrm{36}}{\mathrm{81}}\left(\mathrm{10}^{{n}} −\mathrm{1}\right)^{\mathrm{2}} +\frac{\mathrm{8}}{\mathrm{9}}\left(\mathrm{10}^{{n}} −\mathrm{1}\right) \\ $$$$\left\{\frac{\mathrm{4}}{\mathrm{9}}\left(\mathrm{10}^{{n}} −\mathrm{1}\right)^{\mathrm{2}} \right\}+\frac{\mathrm{8}}{\mathrm{9}}\left(\mathrm{10}^{{n}} −\mathrm{1}\right) \\ $$$$\frac{\mathrm{4}}{\mathrm{9}}\left(\mathrm{10}^{{n}} −\mathrm{1}\right)\left(\mathrm{10}^{{n}} −\mathrm{1}+\mathrm{2}\right) \\ $$$$\frac{\mathrm{4}}{\mathrm{9}}\left(\mathrm{10}^{\mathrm{2}{n}} −\mathrm{1}\right) \\ $$$$\mathrm{4}\left(\frac{\mathrm{10}^{\mathrm{2}{n}} −\mathrm{1}}{\mathrm{10}−\mathrm{1}}\right) \\ $$$$\left[{a}\left(\frac{{r}^{{n}} −\mathrm{1}}{{r}−\mathrm{1}}\right)={a}+{ar}+{ar}^{\mathrm{2}} +...+{ar}^{{n}} \right] \\ $$$$\mathrm{4}\left(\mathrm{1}+\mathrm{10}+\mathrm{10}^{\mathrm{2}} +\mathrm{10}^{\mathrm{3}} +...+\mathrm{10}^{\mathrm{2}{n}} \right) \\ $$$$\left(\mathrm{4}+\mathrm{40}+\mathrm{400}+\mathrm{4000}+.....\right) \\ $$$$\left(\mathrm{444}....\mathrm{4}\right)\leftarrow\mathrm{2}{n}\:{digit} \\ $$ | ||
Commented by mr W last updated on 16/Mar/19 | ||
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$${good}\:{job}\:{sir}! \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 16/Mar/19 | ||
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$${thsnk}\:{you}\:{sir}... \\ $$ | ||