Question Number 220973 by MrGaster last updated on 21/May/25
![Prove:Σ_(k=0) ^n (−(1/3))^k cos^3 (3^(k−n) π)=(3/4)[(−(1/3))^(n+1) +cos(π/3^n )]](https://www.tinkutara.com/question/Q220973.png)
$$\mathrm{Prove}:\underset{{k}=\mathrm{0}} {\overset{{n}} {\sum}}\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{k}} \mathrm{cos}^{\mathrm{3}} \left(\mathrm{3}^{{k}−{n}} \pi\right)=\frac{\mathrm{3}}{\mathrm{4}}\left[\left(−\frac{\mathrm{1}}{\mathrm{3}}\right)^{{n}+\mathrm{1}} +\mathrm{cos}\frac{\pi}{\mathrm{3}^{{n}} }\right] \\ $$
Answered by universe last updated on 22/May/25
![we know that cos^3 x = (1/4)[cos3x +3cosx] put x = (3^k /3^n ) (((−1)/3))^k cos^3 (3^(k−n) π) = (1/4)[(((−1)/3))^k cos(3^(k−n+1) π)+ (((−1)/3))^(k−1) cos(3^(k−n) π) S = (1/4)Σ_(k=1) ^n [(((−1)/3))^k cos(3^(k−n+1) π)+ (((−1)/3))^(k−1) cos(3^(k−n) π)]_(telescoping) S = (1/4)[(((−1)/3))^n (−1)+(((−1)/3))^(−1) cos(3^(−n) π)] S = (3/4)[(((−1)/3))^(n+1) +cos((π/3^n ))]](https://www.tinkutara.com/question/Q221009.png)
$${we}\:{know}\:{that}\:\mathrm{cos}^{\mathrm{3}} {x}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\left[\mathrm{cos3}{x}\:+\mathrm{3cos}{x}\right] \\ $$$$\:{put}\:{x}\:=\:\frac{\mathrm{3}^{{k}} }{\mathrm{3}^{{n}} } \\ $$$$\:\:\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{k}} \mathrm{cos}^{\mathrm{3}} \left(\mathrm{3}^{{k}−{n}} \pi\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{k}} \mathrm{cos}\left(\mathrm{3}^{{k}−{n}+\mathrm{1}} \pi\right)+\:\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{k}−\mathrm{1}} \mathrm{cos}\left(\mathrm{3}^{{k}−{n}} \pi\right)\right. \\ $$$$\:\:{S}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\underset{{telescoping}} {\underbrace{\left[\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{k}} \mathrm{cos}\left(\mathrm{3}^{{k}−{n}+\mathrm{1}} \pi\right)+\:\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{k}−\mathrm{1}} \mathrm{cos}\left(\mathrm{3}^{{k}−{n}} \pi\right)\right]}}\: \\ $$$$\:{S}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\left[\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{n}} \left(−\mathrm{1}\right)+\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{−\mathrm{1}} \mathrm{cos}\left(\mathrm{3}^{−{n}} \pi\right)\right] \\ $$$$\:{S}\:=\:\frac{\mathrm{3}}{\mathrm{4}}\left[\left(\frac{−\mathrm{1}}{\mathrm{3}}\right)^{{n}+\mathrm{1}} +\mathrm{cos}\left(\frac{\pi}{\mathrm{3}^{{n}} }\right)\right] \\ $$$$ \\ $$