Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. $$a_{0}=-1, a_{1}=5, a_{k}=a_{k-2}+a_{k-1}$$

Step-by-Step Solution

Answer

The first five terms of the sequence are -1, 5, 4, 9, 13.

Step 1: Identify and Write the Given Terms

The first two terms of the sequence are already given as \(a_{0} = -1\) and \(a_{1} = 5\)

Step 2: Apply the Recursive Rule to Find the Third Term

To find the third term, apply the given recursive rule, which states that each term \(a_{k}\) is the sum of its two preceding terms, \(a_{k-1}\) and \(a_{k-2}\). So, \(a_{2}=a_{1}+a_{0}\), that means \(a_{2}= 5 + (-1)= 4\)

Step 3: Apply the Recursive Rule to Find the Fourth Term

Repeat Step 2 for the other terms. The fourth term \(a_{3}\) is the sum of the third and second terms. So, \(a_{3}=a_{2}+a_{1}\), that means \(a_{3}= 4 + 5= 9\)

Step 4: Apply the Recursive Rule to Find the Fifth Term

Continuing the process, the fifth term \(a_{4}\) is the sum of the fourth and third terms. So, \(a_{4}=a_{3}+a_{2}\), that means \(a_{4}= 9 + 4= 13\)

Step 5: Write Down All Found Terms

After applying the rules and calculations, the first five terms of the sequence are found to be, \(a_{0}=-1\), \(a_{1}=5\), \(a_{2}=4\), \(a_{3}=9\), and \(a_{4}=13\)

Key Concepts

Sequence Terms

A sequence is essentially a list of numbers, where each number is referred to as a 'term.' In the context of our exercise, sequence terms are the individual numbers that make up our list, defined by following a specific rule. In this particular example, we are dealing with a sequence defined by the initial terms and a recursive rule. The sequence starts with \(a_0 = -1\) and \(a_1 = 5\). From here, each subsequent term is calculated based on the ones before it. This forms an ordered list of numbers, which in our case starts off as -1, 5, and is extended further using the given rule.Knowing the sequence's starting values is crucial. These initial terms allow us to kick off the process of determining all other terms in the sequence.

Recursive Rule

A recursive rule tells us how to generate terms in a sequence based on the preceding ones. This rule acts like a formula or a set of instructions to find the next number, making it different from a formula that gives the \(n^{th}\) term directly.In this exercise, the recursive rule is: \(a_{k} = a_{k-2} + a_{k-1}\). This equation means each term in the sequence is the sum of the previous two terms. For example:

• The second term is the sum of the first and zeroth terms: \(a_2 = a_1 + a_0 = 5 + (-1) = 4\).
• The third term builds upon the first and second: \(a_3 = a_2 + a_1 = 4 + 5 = 9\).
• And it goes on similarly with \(a_4 = a_3 + a_2 = 9 + 4 = 13\).

The recursive rule continuously uses previously calculated terms to generate the next one, illustrating a chain-like process that relies on known values.

Arithmetic Calculations

Arithmetic calculations are crucial in the process of applying a recursive rule. They require basic operations like addition or subtraction to find new terms of the sequence.In the exercise, the arithmetic steps are straightforward. Each requires adding two numbers:

• Start with \(a_0\) and \(a_1\), the initial known values.
• Determine \(a_2\) by performing \(5 + (-1) = 4\).
• The calculation of \(a_3\) involves \(4 + 5 = 9\).
• Finally, find \(a_4\) using \(9 + 4 = 13\).

Careful arithmetic ensures the accuracy of each term, as errors can propagate. By breaking down the calculations, we clearly see how each term's derivation is supported by basic mathematical operations. This systematic approach helps in understanding and accurately generating sequence terms.