Solution to Define an by a0 = 1, a1 = 2, a2 = 4 and an+2 = … - Sikademy

Nov. 27, 2020

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Archangel Macsika

Define an by a0 = 1, a1 = 2, a2 = 4 and an+2 = an+1 + an + an−1, for n ≥ 1. Show that an ≤ 2n for all n ∈ N.

Solution for Define an by a0 = 1, a1 = 2, a2 = 4 and an+2 = an+1 + an + an-1, for n ≥ 1. Show that an ≤ 2n for all n ∈ N.

We use induction on n. The inequality is true for n = 0, 1 and 2. Suppose that it is true for all n ≤ k where k ≥ 2.

Then,

ak+1 = ak + ak-1 + ak-2 ≤ 2k + 2k-1 + 2k-2

= 7 · 2k-2 < 2k+1

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