Question

The Fibonacci sequence is defined by a₁ = 1 = a₂, a_n = a_n–1 + a_n–2 for n > 2. Find (a_n+1)/a_n for n = 1, 2, 3, 4, 5.

Answer 1

Given as

a₁ = 1

a₂ = 1

a_n = a_n–1 + a_n–2

Now, when n = 1:

(a_n+1)/a_n = (a₁₊₁)/a₁

= a₂/a₁

= 1/1

= 1

a₃ = a_3–1 + a_3–2

= a₂ + a₁

= 1 + 1

= 2

When n = 2:

(a_n+1)/a_n = (a₂₊₁)/a₂

= a₃/a₂

= 2/1

= 2

a₄ = a_4–1 + a_4–2

= a₃ + a₂

= 2 + 1

= 3

When n = 3:

(a_n+1)/a_n = (a₃₊₁)/a₃

= a₄/a₃

= 3/2

a₅ = a_5–1 + a_5–2

= a₄ + a₃

= 3 + 2

= 5

When n = 4:

(a_n+1)/a_n = (a₄₊₁)/a₄

= a₅/a₄

= 5/3

a₆ = a_6–1 + a_6–2

= a₅ + a₄

= 5 + 3

= 8

When n = 5:

(a_n+1)/a_n = (a₅₊₁)/a₅

= a₆/a₅ = 8/5

Hence, value of (a_n+1)/a_n when n = 1, 2, 3, 4, 5 are 1, 2, 3/2, 5/3, 8/5