The Fibonacci numbers are a sequence numbers defined by F₁ = 1, F₂Fn = Fn-1 + Fn-2 for n> 3.(a) List the first ten Fibonacci numbers.(b) Show that the sum of the first n Fibonacci numbers equals Fn+2 — 1.=1, and

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Advanced Engineering Mathematics

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Author:Erwin Kreyszig

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The Fibonacci numbers are a sequence numbers defined by F₁ = 1, F₂
Fn = Fn-1 + Fn-2 for n> 3.
(a) List the first ten Fibonacci numbers.
(b) Show that the sum of the first n Fibonacci numbers equals Fn+2 — 1.
=
1, and

Expert Solution

Step 1: First 10 Fibonacci number

Since, the Fibonacci number is defined as $F_{1} = 1, F_{2} = 1$ and $F_{n} = F_{n - 1} + F_{n - 2}$ .

Therefore,

$F_{3} = F_{2} + F_{1} = 1 + 1 = 2$

$F_{4} = F_{3} + F_{2} = 2 + 1 = 3$

$F_{5} = F_{4} + F_{3} = 3 + 2 = 5$

$F_{6} = F_{5} + F_{4} = 5 + 3 = 8$

$F_{7} = F_{6} + F_{5} = 8 + 5 = 13$

$F_{8} = F_{7} + F_{6} = 13 + 8 = 21$

$F_{9} = F_{8} + F_{7} = 21 + 13 = 34$

$F_{10} = F_{9} + F_{8} = 34 + 21 = 55$

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The Fibonacci numbers are a sequence numbers defined by F₁ = 1, F₂ = 1, and Fn = Fn-1 + Fn-2 for n ≥ 3. (a) List the first ten Fibonacci numbers. (b) Show that the sum of the first n Fibonacci numbers equals Fn+2 1.