Question

Consider the sequence defined by aₙ₊₁ = √2+aₙ, a₀=√2, for n= 0, 1, 2, 3, ....

Evaluate the first four terms of the sequence {aₙ}. State the exact values first, and then the approximate values.
a₀= ___ , a₁= ___ , a₂= ____, a₃= ____ (Type exact answers, using radicals as needed.)

Answer 1

The first four terms of the sequence {aₙ} are a₀ = √2, a₁ = 2√2, a₂ = 3√2, a₃ = 4√2. Approximate values are a₀ ≈ 1.414, a₁ ≈ 2.828, a₂ ≈ 4.243, a₃ ≈ 5.657.

Step-by-step explanation:

The sequence {aₙ} is defined as aₙ₊₁ = √2+aₙ, with a₀=√2. To find the first four terms of the sequence, we can apply the recursive formula:

a₀ = √2

a₁ = √2 + a₀ = √2 + √2 = 2√2

a₂ = √2 + a₁ = √2 + 2√2 = 3√2

a₃ = √2 + a₂ = √2 + 3√2 = 4√2

So the exact values of the first four terms of the sequence are a₀ = √2, a₁ = 2√2, a₂ = 3√2, a₃ = 4√2.

To find the approximate values, you can use a calculator to evaluate the square roots. For example, √2 ≈ 1.414. Therefore, the approximate values are a₀ ≈ 1.414, a₁ ≈ 2.828, a₂ ≈ 4.243, a₃ ≈ 5.657.