Question

If p=3+1/p then show p^4=119-1/p^4

Answer 1

Given,

$\displaystyle\sf{\longrightarrow p=3+\dfrac {1}{p}}$

Subtracting $\displaystyle\sf {\dfrac {1}{p},}$

$\displaystyle\sf{\longrightarrow p-\dfrac {1}{p}=3}$

Squaring both sides,

$\displaystyle\sf{\longrightarrow \left (p-\dfrac {1}{p}\right)^2=3^2}$

$\displaystyle\sf{\longrightarrow p^2+\dfrac {1}{p^2}-2=9}$

Adding 2,

$\displaystyle\sf{\longrightarrow p^2+\dfrac {1}{p^2}=11}$

Again squaring,

$\displaystyle\sf{\longrightarrow \left (p^2+\dfrac {1}{p^2}\right)=11^2}$

$\displaystyle\sf{\longrightarrow p^4+\dfrac {1}{p^4}+2=121}$

Subtracting 2,

$\displaystyle\sf{\longrightarrow p^4+\dfrac {1}{p^4}=119}$

Subtracting $\displaystyle\sf {\dfrac {1}{p^4},}$

$\displaystyle\sf{\longrightarrow\underline {\underline { p^4=119-\dfrac {1}{p^4}}}}$

Hence Proved!