Question Number 6831 by Tawakalitu. last updated on 30/Jul/16 | ||
$${Find}\:{the}\:\mathrm{100}{th}\:{digits}\:{of}\:{this}\:{sum}\: \\ $$$$\mathrm{1}\:+\:\mathrm{50}\:+\:\mathrm{50}^{\mathrm{2}} \:+\:.....\:+\:\mathrm{50}^{\mathrm{999}} \\ $$ | ||
Commented by prakash jain last updated on 31/Jul/16 | ||
$$\mathrm{How}\:\mathrm{did}\:\mathrm{you}\:\mathrm{get} \\ $$$$\mathrm{5}^{\mathrm{1000}} ×\mathrm{10}^{\mathrm{1000}} −\mathrm{1}=\left(\mathrm{5}^{\mathrm{1000}} −\mathrm{1}\right)×\left(\mathrm{999}...\mathrm{9999}\right)? \\ $$ | ||
Commented by Yozzii last updated on 30/Jul/16 | ||
$$\underset{{i}=\mathrm{0}} {\overset{\mathrm{999}} {\sum}}\mathrm{50}^{{i}} =\frac{\mathrm{50}^{\mathrm{1000}} −\mathrm{1}}{\mathrm{50}−\mathrm{1}}=\frac{\mathrm{50}^{\mathrm{1000}} −\mathrm{1}}{\mathrm{49}} \\ $$$$=\frac{\mathrm{5}^{\mathrm{1000}} ×\mathrm{10}^{\mathrm{1000}} −\mathrm{1}}{\mathrm{49}} \\ $$$${Now},\:{n}×\mathrm{10}^{{m}} ={n}\mathrm{000}...\mathrm{000}\:\:\left({m}\:{zeroes}\right)\:\left({n}\in\mathbb{N}\right) \\ $$$$\Rightarrow{n}\mathrm{10}^{{m}} −\mathrm{1}=\left({n}−\mathrm{1}\right)\mathrm{999}...\mathrm{999}\:\left({m}\:{nines}\right) \\ $$$$ \\ $$$$\therefore\:\mathrm{5}^{\mathrm{1000}} ×\mathrm{10}^{\mathrm{1000}} =\left(\mathrm{5}^{\mathrm{1000}} \right)\mathrm{000}...\mathrm{000}\:\left(\mathrm{1000}\:{zeroes}\:{trailing}\right) \\ $$$$\Rightarrow\mathrm{5}^{\mathrm{1000}} ×\mathrm{10}^{\mathrm{1000}} −\mathrm{1}=\left(\mathrm{5}^{\mathrm{1000}} −\mathrm{1}\right)\mathrm{999}...\mathrm{999}\:\left(\mathrm{1000}\:{nines}\:{trailing}\right) \\ $$$$\left({continue}\right) \\ $$ | ||
Commented by Yozzii last updated on 31/Jul/16 | ||
$${I}\:{was}\:{trying}\:{to}\:{write}\:{the}\:{number} \\ $$$${in}\:{decimal}\:{form}. \\ $$$${e}.{g}\:\mathrm{5}^{\mathrm{2}} ×\mathrm{10}^{\mathrm{3}} =\mathrm{25000} \\ $$$$\Rightarrow\mathrm{5}^{\mathrm{2}} ×\mathrm{10}^{\mathrm{3}} −\mathrm{1}=\mathrm{24999}=\left(\mathrm{5}^{\mathrm{2}} −\mathrm{1}\right)\mathrm{999}\:\:\left({in}\:{decimal}\right. \\ $$$$\left.{representation},\:{not}\:{a}\:{product}\right) \\ $$ | ||
Commented by Tawakalitu. last updated on 31/Jul/16 | ||
$${Thanks}\:{sir} \\ $$ | ||
Commented by Tawakalitu. last updated on 31/Jul/16 | ||
$${Thanks}\:{sir} \\ $$ | ||