Hello differentiate Originally Posted by

**differentiate** The number x and the real number $\displaystyle \theta $ are such that $\displaystyle x + \frac{1}{x}= 2 cos\theta $ . Use

mathematical induction to show that $\displaystyle x^n+ \frac{1}{x^n}= 2 cosn \theta $ for all positive integers n > 1.

Thank you

Hint:

Start with:

$\displaystyle \left(x^k+\dfrac1{x^k}\right)\left(x+\dfrac1x\righ t)=x^{k+1}+\dfrac{1}{x^{k+1}}+x^{k-1}+\dfrac{1}{x^{k-1}}$

$\displaystyle \Rightarrow x^{k+1}+\dfrac{1}{x^{k+1}}=\left(x^k+\dfrac1{x^k}\ right)\left(x+\dfrac1x\right)-\left(x^{k-1}+\dfrac{1}{x^{k-1}\right)}$

Then assume that the result is true for all values up to $\displaystyle n = k$, and the RHS becomes $\displaystyle 2(2\cos n\theta\cos\theta - \cos(n-1)\theta)$, which simplifies to $\displaystyle 2\cos(n+1)\theta$.

Can you complete the proof?

Grandad