Question

What is the next term in the following sequence? $$ 1,1,2,3,5,8,13,21, \ldots $$

Short answer

The next term in the Fibonacci sequence \(1, 1, 2, 3, 5, 8, 13, 21, \ldots\) is 34, obtained by adding the last two terms: \(13 + 21 = 34\).

Step-by-step solution

Identify the sequence pattern

To find the next term in the sequence, we need to first identify the pattern. The Fibonacci sequence is generated by adding the two previous numbers to get the next number in the sequence. So, the first two numbers are 1 and 1, and the subsequent numbers are obtained by adding the previous two numbers.

Apply the Fibonacci pattern

Applying the Fibonacci pattern to the given sequence, we can deduce the next numbers: - \(1 + 1 = 2\) - \(1 + 2 = 3\) - \(2 + 3 = 5\) - \(3 + 5 = 8\) - \(5 + 8 = 13\) - \(8 + 13 = 21\) Now we can find the next number in the sequence using the same pattern.

Calculate the next term in the sequence

To find the next term, we add the last two terms in the given sequence: \(13 + 21 = 34\) So, the next term in the sequence is 34.

Key concepts

Sequence Patterns

Sequence patterns form the backbone of various mathematical sequences, representing the specific rules or formulas that define the arrangement of elements within a sequence. In the context of the Fibonacci sequence—the focus of our textbook exercise—the pattern is based on the principle of addition. Each term in the sequence is the sum of the two preceding terms.

For students who find it challenging to identify patterns, it's important to look for repetition or a discernable relationship among the numbers. In the Fibonacci sequence, this relationship starts with two 1s and continues indefinitely. By observing the sequence, we see that after the initial two 1s, each subsequent term increases in value. This increase occurs because it's the sum of the two previous numbers, illustrating the inherent pattern in the sequence: - 1 + 1 = 2- 1 + 2 = 3- 2 + 3 = 5

Once the pattern is understood, predicting future terms becomes straightforward, and we can use this concept to solve similar problems involving sequence patterns.

Mathematical Sequences

Mathematical sequences are ordered lists of numbers that follow a specific rule or set of rules. Sequences like the Fibonacci series, featured in our exercise, are just one example of an infinite variety of sequences that can be defined by mathematicians. Sequences can be as simple as a string of consecutive numbers, like 1, 2, 3, 4, 5, ..., or more complex, involving algebraic expressions for their formation.

When dealing with mathematical sequences, recognizing the rule governing the sequence is crucial. The rule for the Fibonacci sequence is relatively simple and yet generates a pattern that has fascinated mathematicians for centuries. It is essential for students to understand that the defining rule of a sequence should be applicable to all its terms, and sometimes multiple rules can generate the same sequence. This can lead to deeper insights into mathematical properties and their applications.

Arithmetic Sequences

Arithmetic sequences are a fundamental type of mathematical sequence where the difference between consecutive terms is constant. Unlike the Fibonacci sequence, which adds the previous two numbers, an arithmetic sequence might simply add (or subtract) the same value each time. This constant value is known as the common difference, denoted by 'd'. For instance, in the sequence 2, 5, 8, 11, ..., each number increases by 3—so, the common difference is 3.

Understanding arithmetic sequences is helpful as they are often used in real-life scenarios, such as calculating the total number of seats in an auditorium if each row has a fixed number of additional seats than the previous one. While the Fibonacci sequence is not an arithmetic sequence, grasping the concept of arithmetic sequences can aid students in recognizing and working with all types of sequences, including those that are more complex. It's the simplicity and predictability of arithmetic sequences that make them a good starting point for students beginning to explore the wide world of mathematical sequences.