Discussion

Question:

It is given that the sequence {x_n} satisfies x₁ = 0, x_n+1 = x_n + 1 + $2 \sqrt{1 + x_{n}}$ for n = 1,2, . . . . . Then x₃₁ is

Explanation:

x₁ = 0 = 1² - 1

x₂ = x₁ + 1 + $2 \sqrt{1 + x_{0}}$ = 0 + 1 + $2 \sqrt{1 + 0}$ = 3 = 2² - 1

x₃ = x₂ + 1 + $2 \sqrt{1 + x_{1}}$ = 3 + 1 + $2 \sqrt{1 + 3}$ = 8 = 3² - 1

∴ x_n = n² - 1

⇒ x₃₁ = 31² - 1 = 960

Hence, 960.

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