How do you use Binet's formula to find the 4th term of the Fibonacci sequence.

Question

Using Pascal's Triangle and algebra - no calculator.

 

 

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Answer 1

Fibonacci Sequence:  a₁ = 1, a₂ = 1, a_n = a_n-1 + a_n-2, for n≥3.  In other words, the nth term (for n≥3) is the sum of the two preceding terms).  

 

So, the first few terms are 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ...

 

Binet's Formula:  a_n = [(1+√5)ⁿ - (1-√5)ⁿ] / [2ⁿ√5]

 

So, a₄ = [(1+√5)⁴ - (1-√5)⁴] / (16√5)

 

          = { [(1+√5)²]² - [(1-√5)²]² } / (16√5)

 

          = [(6+2√5)² - (6-2√5)²] / (16√5)

 

          = [36 + 24√5 + 20 - (36 - 24√5 + 20)] / (16√5)

 

          = 48√5/(16√5) = 3