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 8468 by FilupSmith last updated on 12/Oct/16

Prove or disprove that: (2k+1)^n ∈O ∀k,n∈Z

$$\mathrm{Prove}\:\mathrm{or}\:\mathrm{disprove}\:\mathrm{that}: \\ $$$$\left(\mathrm{2}{k}+\mathrm{1}\right)^{{n}} \in\mathbb{O}\:\:\:\:\:\:\forall{k},{n}\in\mathbb{Z} \\ $$

Answered by Rasheed Soomro last updated on 12/Oct/16

(2k+1)^n = ((n),(0) )(2k)^n + ((n),(1) )(2k)^(n−1) +...+ ((n),(r) )(2k)^(n−r) + ....+ ((( n)),((n−1)) )(2k)+1 =(2k)( ((n),(0) )(2k)^(n−1) + ((n),(1) )(2k)^(n−2) +...+ ((n),(r) )(2k)^(n−r−1) + ....+ ((( n)),((n−1)) ) )+1 =2km+1∈O Hence (2k+1)^n ∈O Proved.

$$\left(\mathrm{2k}+\mathrm{1}\right)^{\mathrm{n}} =\begin{pmatrix}{\mathrm{n}}\\{\mathrm{0}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}} +\begin{pmatrix}{\mathrm{n}}\\{\mathrm{1}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}−\mathrm{1}} +...+\begin{pmatrix}{\mathrm{n}}\\{\mathrm{r}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}−\mathrm{r}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:....+\begin{pmatrix}{\:\:\:\:\mathrm{n}}\\{\mathrm{n}−\mathrm{1}}\end{pmatrix}\left(\mathrm{2k}\right)+\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\left(\mathrm{2k}\right)\left(\begin{pmatrix}{\mathrm{n}}\\{\mathrm{0}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}−\mathrm{1}} +\begin{pmatrix}{\mathrm{n}}\\{\mathrm{1}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}−\mathrm{2}} +...+\begin{pmatrix}{\mathrm{n}}\\{\mathrm{r}}\end{pmatrix}\left(\mathrm{2k}\right)^{\mathrm{n}−\mathrm{r}−\mathrm{1}} \right. \\ $$$$\left.\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:+\:....+\begin{pmatrix}{\:\:\:\:\mathrm{n}}\\{\mathrm{n}−\mathrm{1}}\end{pmatrix}\:\right)+\mathrm{1} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:=\mathrm{2km}+\mathrm{1}\in\mathbb{O} \\ $$$$\mathrm{Hence}\:\left(\mathrm{2k}+\mathrm{1}\right)^{\mathrm{n}} \in\mathbb{O} \\ $$$$\mathrm{Proved}. \\ $$$$ \\ $$

Commented by FilupSmith last updated on 12/Oct/16

Thanks. I was certain this was how to solve this problem!

$$\mathrm{Thanks}.\:\mathrm{I}\:\mathrm{was}\:\mathrm{certain}\:\mathrm{this}\:\mathrm{was}\:\mathrm{how}\:\mathrm{to} \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{problem}! \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com