Question

Write the first five terms of a sequence, don’t make your sequence too simple. Write both an explicit formula and a recursive formula for a general term in the sequence.

Answer 1

One example of a sequence is the Fibonacci sequence, which starts with 0 and 1, and each subsequent term is the sum of the two preceding terms:

0, 1, 1, 2, 3, ...

To write the explicit formula for the nth term of the Fibonacci sequence, we can use Binet's formula:

Fn = [((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n]/sqrt(5)

where Fn is the nth term in the sequence.

To write the recursive formula for the Fibonacci sequence, we can use the definition:

F0 = 0, F1 = 1, and Fn = Fn-1 + Fn-2 for n ≥ 2.

So the first five terms of the Fibonacci sequence are:

F0 = 0

F1 = 1

F2 = 1 (0 + 1)

F3 = 2 (1 + 1)

F4 = 3 (1 + 2)

F5 = 5 (2 + 3)

The explicit formula for the nth term in the sequence is:

Fn = [((1 + sqrt(5))/2)^n - ((1 - sqrt(5))/2)^n]/sqrt(5)

The recursive formula for the nth term in the sequence is:

Fn = Fn-1 + Fn-2 for n ≥ 2, with F0 = 0 and F1 = 1.