Question

If $x+\frac{1}{x}=5$, then the sum of the digits of $x^9+\frac{1}{x^9}$ is

• A. 18
• B. 24
• C. 20
• D. 30

Answer 1

The correct option is C 20

Given: $x+\frac{1}{x}=5$
Cubing both sides, we get:
$(x+\frac{1}{x}{)}^3=5^3$
$\Rightarrow x^3+\frac{1}{x^3}+3x\times \frac{1}{x}(x+\frac{1}{x})=125$

[$(x+y{)}^3=x^3+y^3+3xy(x+y)$]
$\Rightarrow x^3+\frac{1}{x^3}+3(x+\frac{1}{x})=125$
$\Rightarrow (x^3+\frac{1}{x^3})+3\times 5=125$
($\therefore x+\frac{1}{x}=5$)
$\Rightarrow (x^3+\frac{1}{x^3})=110$
Again, cubing both sides, we get:
$(x^3+\frac{1}{x^3}{)}^3=(110{)}^3$
$\Rightarrow x^9+\frac{1}{x^9}+3x^3\times \frac{1}{x^3}(x^3+\frac{1}{x^3})=1331000$
$\Rightarrow x^9+\frac{1}{x^9}+3(x^3+\frac{1}{x^3})=1331000$
$\Rightarrow x^9+\frac{1}{x^9}+3\times 110=1331000$
$(\therefore x^3+\frac{1}{x^3}=110)$
$\Rightarrow x^9+\frac{1}{x^9}=1331000-330=1330670$
∴ Sum of digits = 1 + 3 + 3 + 0 + 6 + 7 + 0 = 20
Hence, the correct answer is option (3).