Menu Close

Category: Number Theory

Find-the-value-of-2-15-2-35-2-63-2-99-2-9999-

Question Number 13121 by tawa tawa last updated on 14/May/17 $$\mathrm{Find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\::\:\:\frac{\mathrm{2}}{\mathrm{15}}\:+\:\frac{\mathrm{2}}{\mathrm{35}}\:+\:\frac{\mathrm{2}}{\mathrm{63}}\:+\:\frac{\mathrm{2}}{\mathrm{99}}\:+\:…\:+\:\frac{\mathrm{2}}{\mathrm{9999}} \\ $$ Answered by nume1114 last updated on 15/May/17 $$\:\:\:\:\frac{\mathrm{2}}{\mathrm{15}}+\frac{\mathrm{2}}{\mathrm{35}}+\frac{\mathrm{2}}{\mathrm{63}}+\frac{\mathrm{2}}{\mathrm{99}}+…+\frac{\mathrm{2}}{\mathrm{9999}} \\ $$$$=\frac{\mathrm{2}}{\mathrm{3}×\mathrm{5}}+\frac{\mathrm{2}}{\mathrm{5}×\mathrm{7}}+\frac{\mathrm{2}}{\mathrm{7}×\mathrm{9}}+…+\frac{\mathrm{2}}{\mathrm{99}×\mathrm{101}} \\ $$$$=\underset{{k}=\mathrm{1}}…

Express-n-1-49-1-n-n-2-1-as-a-b-2-for-some-integers-a-and-b-

Question Number 131484 by bemath last updated on 05/Feb/21 $$\mathrm{Express}\:\underset{\mathrm{n}=\mathrm{1}} {\overset{\mathrm{49}} {\sum}}\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{n}+\sqrt{\mathrm{n}^{\mathrm{2}} −\mathrm{1}}}}\:\mathrm{as}\:{a}+{b}\sqrt{\mathrm{2}} \\ $$$$\mathrm{for}\:\mathrm{some}\:\mathrm{integers}\:{a}\:\mathrm{and}\:{b} \\ $$ Answered by benjo_mathlover last updated on 05/Feb/21 Answered…

1-3-2005-have-how-many-digits-in-periodic-part-

Question Number 403 by 123456 last updated on 25/Jan/15 $$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{2005}} }\:\mathrm{have}\:\mathrm{how}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{periodic}\:\mathrm{part}? \\ $$ Answered by prakash jain last updated on 30/Dec/14 $$\mathrm{3}^{\mathrm{2003}} \:\mathrm{digits}\:\mathrm{will}\:\mathrm{be}\:\mathrm{present}\:\mathrm{in}\:\mathrm{periodic}\:\mathrm{part}. \\ $$$$\mathrm{10}^{{n}}…

1-3-20-have-how-many-digits-in-periodic-part-

Question Number 400 by 123456 last updated on 25/Jan/15 $$\frac{\mathrm{1}}{\mathrm{3}^{\mathrm{20}} }\:\mathrm{have}\:\mathrm{how}\:\mathrm{many}\:\mathrm{digits}\:\mathrm{in}\:\mathrm{periodic}\:\mathrm{part}? \\ $$ Answered by prakash jain last updated on 29/Dec/14 $$\mathrm{3}^{\mathrm{20}} \:\mathrm{is}\:\mathrm{not}\:\mathrm{divisible}\:\mathrm{by}\:\mathrm{2}\:\mathrm{or}\:\mathrm{5}.\:\mathrm{So} \\ $$$$\mathrm{periodicity}\:\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\mathrm{equation}…

a-n-1-2-3-4-5-n-n-1-mod-2-a-n-1-2-3-4-5-6-n-n-0-mod-2-lim-n-a-n-1-

Question Number 348 by 123456 last updated on 23/Dec/14 $${a}_{{n}} =\mathrm{1}−\frac{\mathrm{2}}{\mathrm{3}−\frac{\mathrm{4}}{\mathrm{5}−{n}}},{n}\equiv\mathrm{1}\left(\mathrm{mod}\:\mathrm{2}\right) \\ $$$${a}_{{n}} =\mathrm{1}−\frac{\mathrm{2}}{\mathrm{3}−\frac{\mathrm{4}}{\mathrm{5}−\frac{\mathrm{6}}{{n}}}},{n}\equiv\mathrm{0}\left(\mathrm{mod}\:\mathrm{2}\right) \\ $$$$\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:{a}_{{n}} \overset{?} {=}\mathrm{1} \\ $$ Commented by 123456 last…