Question

If x⁵ + 1/x⁵ = 2525, then x + 1/x = 

(a) 4

(b) 5

(c) 6

(d) 7

Answer 1

Correct answer is:- (b) 5

Explanation:- 

Let \(x\) + \(\frac{1}{x} \) be y

\(x^5\) + \(\frac{1}{x^5} \)= (x² + 1/x²) (x³ + 1/x³) - (x + 1/x)

\(x^5\) + \(\frac{1}{x^5} \) = (y² - 2) (y³ - 3y) - y 

Now, 

(y² - 2) (y³ - 3y) - y  = 2525

y⁵ - 5y³ + 5y = 2525

Take out the smallest factor of each term (excluding 1) 

y(y⁴ - 5y² + 5) = 5 × 505

By comparing, 

 y = 5 … (1) 

Taking 2nd part, we get 

⇒ y⁴ - 5y² + 5 = 505

⇒ y⁴ - 5y² - 500 = 0

⇒ x² - 5x - 500  = 0 (Taking y² = x) 

⇒ (x - 25) (x + 20) = 0

⇒ x = 25 (x = -20 not possible) 

⇒ y² = 25

∴ y = ±5 ... (2)

From (1) and (2) 

y = 5

Therefore, \(x\) + \(\frac{1}{x} \)  = 5.