Question

please help thank you x

Answer 1

We know that :

A Geometric Sequence is in the form of :

a , ar , ar², ar³. . . . . .arⁿ⁻¹ (where a is the first term and r is the common ratio)

Given that √x + 1 , 1 and √x - 1 are the 1st three terms of Geometric Sequence

⇒ a = √x + 1

⇒ ar = 1

⇒ ar² = √x - 1

We know that ar/a = ar²/ar = r

substituting the respective values we get :

⇒ 1/(√x + 1) = (√x - 1)/1

⇒ (√x)² - 1 = 1

⇒ x = 1 + 1

⇒ x = 2

(b) We know that nth term of a Geometric Sequence is Given by arⁿ⁻¹

⇒ 5th term of the Geometric sequence is given by ar⁵⁻¹ = ar⁴

We know that a = √2 + 1 and r = 1/(√2 + 1)

substituting we get :

5th term = √2 + 1 × 1/(√2 + 1)⁴

5th term = 1/(√2 + 1)³

We know that (√2 + 1)³ = 2√2 + 1 + 6 + 3√2 = 7 + 5√2

5th term = 1/(7 + 5√2)

Rationalizing with 5√2 - 7 we get :

5th term = 5√2 - 7

Comparing with a√2 + b we can say that a = 5 and b = -7