Fibonacci Sequence Series Completion: Finding the Missing Number

This question asks us to find the next number in the given numerical sequence: 0, 1, 1, 2, 3, 5, 8, ?. To solve this, we need to identify the underlying pattern or rule governing the series.

Identifying the Series Pattern

Let's examine the relationship between consecutive numbers in the series:

• The first number is 0.
• The second number is 1.
• The third number is 1, which is the sum of the first two numbers ($0 + 1 = 1$).
• The fourth number is 2, which is the sum of the second and third numbers ($1 + 1 = 2$).
• The fifth number is 3, which is the sum of the third and fourth numbers ($1 + 2 = 3$).
• The sixth number is 5, which is the sum of the fourth and fifth numbers ($2 + 3 = 5$).
• The seventh number is 8, which is the sum of the fifth and sixth numbers ($3 + 5 = 8$).

The pattern observed is that each number (starting from the third number) is the sum of the two preceding numbers. This is known as the Fibonacci sequence.

Mathematical Representation

The Fibonacci sequence can be defined recursively as:

Let $F_n$ denote the $n^{th}$ term in the sequence.

• $F_0 = 0$
• $F_1 = 1$
• $F_n = F_{n-1} + F_{n-2}$ for $n > 1$

Calculating the Missing Term

We need to find the term that follows 8. Based on the identified pattern, this term will be the sum of the last two known terms (5 and 8).

The last two terms are $F_6 = 8$ and $F_5 = 5$.

The missing term is $F_7$:

$F_7 = F_6 + F_5$

$F_7 = 8 + 5$

$F_7 = 13$

Therefore, the completed series is 0, 1, 1, 2, 3, 5, 8, 13.