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

Given $y+\frac{1}{y}=3$ ..................(1)

square both sides

Then ${(y+\frac{1}{y})}^2=(3{)}^2$

$\Rightarrow y^2+\frac{1}{y^2}+2=9$

$\Rightarrow y^2+\frac{1}{y^2}=9-2$

$\Rightarrow y^2+\frac{1}{y^2}=7$................(2)

cube both sides of (1)

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

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

$\Rightarrow y^3+\frac{1}{y^3}=27-9$

$\Rightarrow y^3+\frac{1}{y^3}=18$...................(3)

multiply (2) and (3)

We get

$(y^3+\frac{1}{y^3})(y^2+\frac{1}{y^2})=18\times 7$

$\Rightarrow y^5+\frac{1}{y^5}+(y+\frac{1}{y})=126$

$\Rightarrow y^5+\frac{1}{y^5}+3=126$

$\Rightarrow y^5+\frac{1}{y^5}=126-3=123$