Question

The Fibonacci sequence is defined by 1 = a₁ = a₂ and an = a_{n – 1} + a_{n – 2}, n > 2.

Find `a_(n+1)/a_n`, for n = 1, 2, 3, 4, 5

Answer 1

a₁ = 1, a₂ = 1

a_n = a_n−1 + a_n−2

Putting n = 3, 4, 5, 6,

a₃ = a₂ + a₁ = 1 + 1 = 2

a₄ = a₃ + a₂ = 2 + 1 = 3

a₅ = a₄ + a₃ = 3 + 2 = 5

a₆ = a₅ + a₄ = 5 + 3 = 8

Now, substituting n = 1, 2, 3, 4, 5 in `"a"_("n" + 1)/"a"_"n"`

∴ n = `1, (a_n + 1)/(a_n) = a_2/a_1 = 1/1 = 1`,

n = `2, (a_n + 1)/(a_n) = "a"_3/"a"_2 = 2/1 = 2`,

n = `3, (a_n + 1)/(a_n) = "a"_4/"a"_3 = 3/2`,

n = `4, (a_n + 1)/(a_n) = "a"_5/"a"_4 = 5/3`,

n = `5, (a_n + 1)/(a_n) = "a"_6/"a"_5 = 8/5`