Category: Arithmetic

Question Number 191821 by mehdee42 last updated on 01/May/23 $$Q▸Showthat:$$$$\underset{i=1}{\sum ^{2n}}{(-1)}^{i+1}\frac{1}{i}=\underset{i=1}{\sum ^n}\frac{1}{i+n}$$ Answered by mehdee42 last updated on…

Question Number 60441 by ajfour last updated on 21/May/19 $$\mathrm{If}\mathrm{a}\mathrm{sum}\mathrm{of}\mathrm{money}\mathrm{doubles}\mathrm{itself}$$$$\mathrm{in}\mathrm{a}\mathrm{time}\mathrm{T},\mathrm{when}\mathrm{compounded}$$$$\mathrm{continuously},\mathrm{find}\mathrm{the}\mathrm{rate}\mathrm{of}$$$$\mathrm{interest},\mathrm{in}\mathrm{terms}\mathrm{of}\mathrm{T}.$$ Answered by tanmay last updated on 21/May/19…

Question Number 125918 by I want to learn more last updated on 15/Dec/20 $$\underset{\mathrm{n}=-\mathrm{\infty }}{\sum ^{\mathrm{\infty }}}\frac{1}{{(\mathrm{n}+\frac{1}{3})}^3}$$ Answered by mindispower last updated on…

Question Number 60155 by Tawa1 last updated on 18/May/19 $$\mathrm{Show}\mathrm{that}:$$$$\frac{1}{2}+\cos \left(\mathrm{x}\right)+\cos \left(2\mathrm{x}\right)+\dots +\cos \left(\mathrm{nx}\right)=\frac{\sin (\mathrm{n}+\frac{1}{2})\mathrm{x}}{2\sin \left(\frac{1}{2}\right)\mathrm{x}}$$$$\mathrm{By}\mathrm{principle}\mathrm{of}\mathrm{mathematical}\mathrm{induction}$$ Answered by Kunal12588 last updated on 18/May/19 $$\:{P}\left({n}\right):\frac{\mathrm{1}}{\mathrm{2}}\:+\:\mathrm{cos}\left(\mathrm{x}\right)\:+\:\mathrm{cos}\left(\mathrm{2x}\right)\:+\:…\:+\:\mathrm{cos}\left(\mathrm{nx}\right)\:\:=\:\:\frac{\mathrm{sin}\left(\mathrm{n}\:+\:\frac{\mathrm{1}}{\mathrm{2}}\right)\mathrm{x}}{\mathrm{2}\:\mathrm{sin}\left(\frac{\mathrm{1}}{\mathrm{2}}\right)\mathrm{x}} \…

Question Number 191028 by Mingma last updated on 16/Apr/23 Answered by Frix last updated on 16/Apr/23 $$\mathrm{Not}\mathrm{a}\mathrm{proof}\mathrm{but}\mathrm{an}\mathrm{idea}\mathrm{for}\mathrm{a}\mathrm{proof}:$$$$(n-1)n(n+1)\times N=k^3;k\in \mathbb{N}$$$$\mathrm{We}\mathrm{know}\mathrm{that}$$$$\mathrm{1}.\:\mathrm{gcd}\:\left({n}−\mathrm{1},\:{n}\right)\:=\mathrm{1} \…