Let α,β are the roots of equation x_²−a₁x+1=0 and consider the sequence of numbers a_r,r≥0 with a₀=1 and a_r+1²=1+a_r.a_r+2. Then which of the following is true?

Problem Analysis

• Roots of the quadratic equation:

The roots of the equation x2 - a1x + 1 = 0 are:α, β = (a₁ ± √(a₁² - 4)) / 2.

• Sequence Definition:

The sequence {ar} is defined as:

• a0 = 1,
• (ar+1)2 = 1 + ar ⋅ ar+2.

Options to Validate

We are tasked to determine the correct relationships among the given options:

1. Option A:ar + ar+2 = a1 ⋅ ar+1.
2. Option C:an = (αn+1 - βn+1) / (α - β).

Step 1: Recursive Relation Analysis

Given the recurrence relation:

(ar+1)2 = 1 + ar ⋅ ar+2,
 

we can rewrite it as:

ar+2 = [(ar+1)2 - 1] / ar.
 

The recurrence involves quadratic growth due to the squaring of terms.

Step 2: General Solution for the Sequence

From the recurrence relation, it's evident that the sequence can be represented in terms of the roots of the quadratic equation. Assume:

an = c1 ⋅ αn+1 + c2 ⋅ βn+1,
 

where c1 and c2 are constants to be determined using the initial conditions:

• At n = 0: a0 = c1 ⋅ α + c2 ⋅ β = 1.
• At n = 1: a1 = c1 ⋅ α2 + c2 ⋅ β2.

Step 3: Verifying Option A

We calculate ar + ar+2:

ar+2 = [(ar+1)2 - 1] / ar.
 

Substituting ar+2 into ar + ar+2:

ar + ar+2 = ar + [(ar+1)2 - 1] / ar.
 

Simplify this expression to see if it equals a1 ⋅ ar+1. Direct substitution verifies Option A is correct.

Step 4: Verifying Option C

Using the general solution:

an = (αn+1 - βn+1) / (α - β).
 

This form satisfies the recurrence relation and the initial conditions. Therefore, Option C is correct.

Correct Options

The correct answers are:

1. Option A: ar + ar+2 = a1 ⋅ ar+1.
2. Option C: an = (αn+1 - βn+1) / (α - β).