Question

If y+$\frac{1}{y}$=3 then the value of $y^5+\frac{1}{y^5}$ is

• A. $123$
• B. $124$
• C. $125$
• D. $126$

Answer 1

The correct option is A $123$

$\Rightarrow$ $y+\frac{1}{y}=3$

$\Rightarrow$ $(y+\frac{1}{y}{)}^3=(3{)}^3$

$\Rightarrow$ $y^3+3\times y^2\times \frac{1}{y}+3\times y\times \frac{1}{y^2}+\frac{1}{y^3}=27$

$\Rightarrow$ $y^3+\frac{1}{y^3}+3(y+\frac{1}{y})=27$

$\Rightarrow$ $y^3+\frac{1}{y^3}+3\left(3\right)=27$

$\therefore$ $y^3+\frac{1}{y^3}=18$ ----- ( 1 )

$\Rightarrow$ Now, $y+\frac{1}{y}=3$

$\Rightarrow$ $(y+\frac{1}{y}{)}^5=(3{)}^5$

$\Rightarrow$ $y^5+5y^4\left(\frac{1}{y}\right)+10y^3\left(\frac{1}{y^2}\right)+10y^2\left(\frac{1}{y^3}\right)+5y\left(\frac{1}{y^4}\right)+\frac{1}{y^5}=243$

$\Rightarrow$ $y^5+\frac{1}{y^5}+10(y+\frac{1}{y})+5(y^3+\frac{1}{y^3})=243$

$\Rightarrow$ $y^5+\frac{1}{y^5}+10\left(3\right)+5\left(18\right)=243$ [From ( 1 ) ]

$\Rightarrow$ $y^5+\frac{1}{y^5}+30+90=243$

$\Rightarrow$ $y^5+\frac{1}{y^5}=243-120$

$\therefore$ $y^5+\frac{1}{y^5}=123$