Question

If x+1/x= 3, then prove that m^5+1/m^5= 123

Answer 1

9514 1404 393

Step-by-step explanation:

We can start with the relations ...

`\displaystyle\left(x+(1)/(x)\right)^3=\left(x^3+(1)/(x^3)\right)+3\left(x+(1)/(x)\right)\\\\\left(x+(1)/(x)\right)^5=\left(x^5+(1)/(x^5)\right)+5\left(x^3+(1)/(x^3)\right)+10\left(x+(1)/(x)\right)\\\\\textsf{From these, we can derive ...}\\\\x^5+(1)/(x^5)=\left(x+(1)/(x)\right)^5-5\left(\left(x+(1)/(x)\right)^3-3\left(x+(1)/(x)\right)\right)-10\left(x+(1)/(x)\right)`

`\displaystyle x^5+(1)/(x^5)=\left(x+(1)/(x)\right)^5-5\left(x+(1)/(x)\right)^3+5\left(x+(1)/(x)\right)\right)\\\\x^5+(1)/(x^5)=3^5 -5(3^3)+5(3)\\\\=((3^2-5)3^2+5)\cdot3=(4\cdot9+5)\cdot3=(41)(3)\\\\=\boxed{123}`