Problem 17 - Algebra and Trigonometry

Question

\(13-18\) Find the first five terms of the given recursively defined sequence. $$ a_{n}=a_{n-1}+a_{n-2} \quad \text { and } \quad a_{1}=1, a_{2}=2 $$

Step-by-Step Solution

Answer

The first five terms are: 1, 2, 3, 5, 8.

Step 1: Understand the Recursive Formula

The sequence is defined recursively. This means each term is determined by previous terms as given by the formula: \(a_n = a_{n-1} + a_{n-2}\). The first two terms are already provided: \(a_1 = 1\) and \(a_2 = 2\). Our task is to find the next three terms to make a total of five.

Step 2: Compute the Third Term

To find \(a_3\), use the relation \(a_3 = a_2 + a_1\). Substitute the known values: \(a_3 = 2 + 1 = 3\). Thus, \(a_3 = 3\).

Step 3: Compute the Fourth Term

Next, calculate \(a_4\) using the formula: \(a_4 = a_3 + a_2\). Substitute the values we know: \(a_4 = 3 + 2 = 5\). Thus, \(a_4 = 5\).

Step 4: Compute the Fifth Term

Finally, apply the recursive formula to find \(a_5\): \(a_5 = a_4 + a_3\). Substitute the known values: \(a_5 = 5 + 3 = 8\). Thus, \(a_5 = 8\).

Key Concepts

Recursive Formula

A recursive formula is a powerful tool in mathematics used to define a sequence of numbers. It allows us to determine terms in the sequence by using previous terms, rather than starting from scratch each time. The provided sequence uses the recursive formula \(a_n = a_{n-1} + a_{n-2}\) to generate new sequence terms.

This specific formula tells us that each term \(a_n\) is the sum of the two preceding terms.

• \(a_{n-1}\) is the term immediately before \(a_n\).
• \(a_{n-2}\) is the term two places before \(a_n\).

By starting with initial terms given, like \(a_1 = 1\) and \(a_2 = 2\), we can apply the recursive formula to find the next terms in the sequence. Understanding this concept is crucial when dealing with sequences defined recursively, as it provides a straightforward path for calculating multiple terms.

Sequence Terms

Sequence terms are the individual elements or numbers in a sequence. In a recursively defined sequence, each term typically relies on earlier terms. The sequence has a clear order, starting from given initial terms, and progresses by applying the recursive formula.

In our example, the first five terms of the sequence using \(a_1 = 1\) and \(a_2 = 2\) turn out to be:

• \(a_1 = 1\)
• \(a_2 = 2\)
• \(a_3 = 3\) calculated as \(\ 2 + 1\)
• \(a_4 = 5\) calculated as \(\ 3 + 2\)
• \(a_5 = 8\) calculated as \(\ 5 + 3\)

Understanding how each term builds on the previous ones, brings clarity to how sequences increment and forms a solid foundation for sequence-related problems.

Fibonacci-like Sequence

A Fibonacci-like sequence shares properties with the well-known Fibonacci sequence but can have different starting values. This means in place of starting with \(a_1 = 1\) and \(a_2 = 1\), any two numbers can be the initial terms. The rule \(a_n = a_{n-1} + a_{n-2}\) still applies.

Just as in Fibonacci sequences, which start famously with 1 and 1 or sometimes 0 and 1, the rest of the terms follow this pattern of summing the previous two terms. In this case, the starting values are \(1\) and \(2\) instead, producing a unique sequence: 1, 2, 3, 5, and 8 for the first five terms. By exploring these sequences, one can discover numerous patterns and applications, such as in algorithm design and nature's growth patterns, making them an exciting subject to study.