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Category: Number Theory

is-it-correct-when-S-n-1-2-1-3-1-4-1-5-1-n-2-3-2-4-2-5-2-n-3-4-3-5-3-n-4-5-4-n-

Question Number 8139 by sou1618 last updated on 01/Oct/16 $${is}\:{it}\:{correct}? \\ $$$${when} \\ $$$${S}_{{n}} =\left\{\right. \\ $$$$\mathrm{1}×\mathrm{2}+\mathrm{1}×\mathrm{3}+\mathrm{1}×\mathrm{4}+\mathrm{1}×\mathrm{5}+…….+\mathrm{1}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:+\mathrm{2}×\mathrm{3}+\mathrm{2}×\mathrm{4}+\mathrm{2}×\mathrm{5}+…….+\mathrm{2}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{3}×\mathrm{4}+\mathrm{3}×\mathrm{5}+…….+\mathrm{3}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\mathrm{4}×\mathrm{5}+…….+\mathrm{4}×{n} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….…

Given-that-Z-and-H-are-complex-number-obtain-the-real-and-imaginary-of-Z-H-

Question Number 7748 by Tawakalitu. last updated on 13/Sep/16 $${Given}\:{that}\:{Z}\:{and}\:{H}\:{are}\:{complex}\:{number}.\: \\ $$$${obtain}\:{the}\:{real}\:{and}\:{imaginary}\:{of}\:{Z}^{{H}} \\ $$ Answered by Yozzia last updated on 13/Sep/16 $${Let}\:{Z}={re}^{{i}\theta} ,\:{H}={c}+{di}\:\:\left({r},\theta,{c},{d}\in\mathbb{R},\:{r}>\mathrm{0},\:{i}=\sqrt{−\mathrm{1}}\right). \\ $$$${Z}^{{H}}…

w-x-y-z-are-digits-in-respective-base-system-and-a-b-are-bases-Find-out-an-example-examples-which-satisfy-the-following-wxyz-a-wxyz-b-wxyz-a-b-

Question Number 7249 by Rasheed Soomro last updated on 19/Aug/16 $${w},{x},{y},{z}\:{are}\:{digits}\:{in}\:{respective}\:{base}\:{system}\:{and} \\ $$$${a},{b}\:{are}\:{bases}. \\ $$$${Find}\:{out}\:{an}\:{example}/{examples}\:{which}\:{satisfy} \\ $$$${the}\:{following} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{wxyz}_{{a}} +{wxyz}_{{b}} ={wxyz}_{{a}+{b}} \: \\ $$ Commented…

An-Interesting-App-I-ve-recently-downloaded-the-app-called-Math-Tricks-from-the-Google-Play-Store-If-you-d-like-to-improve-your-speed-and-skill-in-the-mental-calculations-arena-I-think-this-app-s

Question Number 7230 by Yozzia last updated on 17/Aug/16 $${An}\:{Interesting}\:{App}: \\ $$$${I}'{ve}\:{recently}\:{downloaded}\:{the}\:{app}\:{called} \\ $$$$'{Math}\:{Tricks}'\:{from}\:{the}\:{Google}\:{Play}\:{Store}. \\ $$$${If}\:{you}'{d}\:{like}\:{to}\:{improve}\:{your}\:{speed} \\ $$$${and}\:{skill}\:{in}\:{the}\:{mental}\:{calculations} \\ $$$${arena},\:{I}\:{think}\:{this}\:{app}\:{should}\:{be}\:{of} \\ $$$${interest}\:{to}\:{you}.\:{The}\:{app}\:{can}\:{throw}\:{you}\:{simple} \\ $$$${calculations}\:{that}\:{include}\:{the}\:{basic}\:{operations} \\…

Let-a-and-b-be-positive-integers-such-that-ab-1-divides-a-2-b-2-Show-that-a-2-b-2-ab-1-is-the-square-of-an-integer-IMO-1988-Qu-6-

Question Number 7213 by Yozzia last updated on 16/Aug/16 $${Let}\:{a}\:{and}\:{b}\:{be}\:{positive}\:{integers}\:{such} \\ $$$${that}\:{ab}+\mathrm{1}\:{divides}\:{a}^{\mathrm{2}} +{b}^{\mathrm{2}} .\:{Show}\:{that} \\ $$$$\frac{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} }{{ab}+\mathrm{1}}\:{is}\:{the}\:{square}\:{of}\:{an}\:{integer}. \\ $$$$\left({IMO}\:\mathrm{1988}\:{Qu}.\mathrm{6}\right) \\ $$ Terms of Service…

Evaluate-sin-3n-n-from-1-to-infinity-

Question Number 7191 by Tawakalitu. last updated on 15/Aug/16 $${Evaluate}\:\:\:\:\:\Sigma\:\frac{{sin}\left(\mathrm{3}{n}\right)}{{n}}\:\:\:\:\:{from}\:\:\mathrm{1}\:\:{to}\:\:{infinity}\: \\ $$ Answered by Yozzia last updated on 15/Aug/16 $${Define}\:{the}\:{function}\: \\ $$$${f}\left({x}\right)={x}\:{for}\:\mathrm{0}<{x}<\mathrm{1}\:,\:{period}=\mathrm{2}. \\ $$$${For}\:{Fourier}\:{series}\:{of}\:{f}\:{having}\:{the}\:{form} \\…

Evaluate-sin-n-n-From-1-to-infinity-

Question Number 7189 by Tawakalitu. last updated on 15/Aug/16 $${Evaluate}\:\:\::\:\:\:\Sigma\:\frac{{sin}\left({n}\right)}{{n}}\:\:,\:\:\:{From}\:\:\:\mathrm{1}\:{to}\:\:{infinity}. \\ $$ Commented by Yozzia last updated on 15/Aug/16 $${f}\left({x}\right)=\frac{{a}_{\mathrm{0}} }{\mathrm{2}}+\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\left\{{a}_{{n}} {cos}\frac{{n}\pi{x}}{{L}}+{b}_{{n}} {sin}\frac{{n}\pi{x}}{{L}}\right\}…

Is-u-n-real-sequence-defende-by-u-0-3e-t-u-n-1-2-u-n-2-1-Determine-the-general-term-u-n-of-this-series-justify-your-answer-and-method-used-

Question Number 7005 by Tawakalitu. last updated on 05/Aug/16 $${Is}\:\:{u}_{{n}} \:{real}\:{sequence}\:{defende}\:\:{by}\:\: \\ $$$${u}_{\mathrm{0}} \:=\:\mathrm{3}{e}^{{t}} \:{u}_{{n}+\mathrm{1}} \:=\:\mathrm{2}\left({u}_{{n}} \right)^{\mathrm{2}} \:−\:\mathrm{1} \\ $$$${Determine}\:{the}\:{general}\:{term}\:{u}_{{n}} \:{of}\:{this}\:{series}\: \\ $$$$\left({justify}\:{your}\:{answer}\:{and}\:{method}\:{used}\right) \\ $$$$…

Question-138003

Question Number 138003 by benjo_mathlover last updated on 09/Apr/21 Answered by EDWIN88 last updated on 09/Apr/21 $${begin}\:{from}\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{x}^{{n}} }{{n}!}\:=\:{e}^{{x}} −\mathrm{1}\:.\:{Taking}\:{derivative} \\ $$$$\Rightarrow\:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{nx}^{{n}−\mathrm{1}}…