Question and Answers Forum

All Questions      Topic List

Algebra Questions

Previous in All Question      Next in All Question      

Previous in Algebra      Next in Algebra      

Question Number 162414 by HongKing last updated on 29/Dec/21

Prove the Identity for any (a,n) in Real Number  (1 + a)∙a^([n])  = a ∙ a^(2[(n/2)])  + a^(2[((n+1)/2)])   [∗] Greatest Integer Function

$$\mathrm{Prove}\:\mathrm{the}\:\mathrm{Identity}\:\mathrm{for}\:\mathrm{any}\:\left(\mathrm{a},\mathrm{n}\right)\:\mathrm{in}\:\mathrm{Real}\:\mathrm{Number} \\ $$$$\left(\mathrm{1}\:+\:\mathrm{a}\right)\centerdot\mathrm{a}^{\left[\boldsymbol{\mathrm{n}}\right]} \:=\:\mathrm{a}\:\centerdot\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}}{\mathrm{2}}\right]} \:+\:\mathrm{a}^{\mathrm{2}\left[\frac{\boldsymbol{\mathrm{n}}+\mathrm{1}}{\mathrm{2}}\right]} \\ $$$$\left[\ast\right]\:\mathrm{Greatest}\:\mathrm{Integer}\:\mathrm{Function} \\ $$

Answered by mindispower last updated on 31/Dec/21

n∈[2k,2k+1[  [n]=2k  (1+a).a^(2k) =a.a^(2k) +a^(2k) =(1+a).a^(2k)   true  n∈[2k−1,2k]  [n]=2k−1  [(n/2)]=k−1  [((n+1)/2)]=k  (1+a).a^(2k−1) =? a.a^(2(k−1)) +a^(2k) =a^(2k−1) +a^(2k) =a^(2k−1) (1+a)  True  (1+a).a^([n]) =a.a^(2[(n/2)]) +a^(2[((n+1)/2)])

$${n}\in\left[\mathrm{2}{k},\mathrm{2}{k}+\mathrm{1}\left[\right.\right. \\ $$$$\left[{n}\right]=\mathrm{2}{k} \\ $$$$\left(\mathrm{1}+{a}\right).{a}^{\mathrm{2}{k}} ={a}.{a}^{\mathrm{2}{k}} +{a}^{\mathrm{2}{k}} =\left(\mathrm{1}+{a}\right).{a}^{\mathrm{2}{k}} \\ $$$${true} \\ $$$${n}\in\left[\mathrm{2}{k}−\mathrm{1},\mathrm{2}{k}\right] \\ $$$$\left[{n}\right]=\mathrm{2}{k}−\mathrm{1} \\ $$$$\left[\frac{{n}}{\mathrm{2}}\right]={k}−\mathrm{1} \\ $$$$\left[\frac{{n}+\mathrm{1}}{\mathrm{2}}\right]={k} \\ $$$$\left(\mathrm{1}+{a}\right).{a}^{\mathrm{2}{k}−\mathrm{1}} =?\:{a}.{a}^{\mathrm{2}\left({k}−\mathrm{1}\right)} +{a}^{\mathrm{2}{k}} ={a}^{\mathrm{2}{k}−\mathrm{1}} +{a}^{\mathrm{2}{k}} ={a}^{\mathrm{2}{k}−\mathrm{1}} \left(\mathrm{1}+{a}\right) \\ $$$${True} \\ $$$$\left(\mathrm{1}+{a}\right).{a}^{\left[{n}\right]} ={a}.{a}^{\mathrm{2}\left[\frac{{n}}{\mathrm{2}}\right]} +{a}^{\mathrm{2}\left[\frac{{n}+\mathrm{1}}{\mathrm{2}}\right]} \\ $$$$ \\ $$

Commented by HongKing last updated on 31/Dec/21

cool my dear Sir thank you so much

$$\mathrm{cool}\:\mathrm{my}\:\mathrm{dear}\:\mathrm{Sir}\:\mathrm{thank}\:\mathrm{you}\:\mathrm{so}\:\mathrm{much} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com