Problem 55 - Algebra and Trigonometry Real Mathematics, Real People

Question

Write the first five terms of the sequence defined recursively. $$a_{1}=3, a_{k+1}=2\left(a_{k}-1\right)$$

Step-by-Step Solution

Answer

The first five terms of the sequence are \(3, 4, 6, 10, 18\)

Step 1: Identify the first term

The first term of the sequence is explicitly given as \(a_{1}=3\)

Step 2: Apply the recursive rule to get the second term

Substitute \(k=1\) into the recursive formula \(a_{k+1}=2\left(a_{k}-1\right)\), we get \(a_{2}=2(a_{1}-1)=2(3-1)=4\)

Step 3: Apply the recursive rule to get the third term

Substitute \(k=2\) into the recursive formula, we get \(a_{3}=2(a_{2}-1)=2(4-1)=6\)

Step 4: Apply the recursive rule to get the fourth term

Substitute \(k=3\) into the recursive formula, we get \(a_{4}=2(a_{3}-1)=2(6-1)=10\)

Step 5: Apply the recursive rule to get the fifth term

Substitute \(k=4\) into the recursive formula, we get \(a_{5}=2(a_{4}-1)=2(10-1)=18\)

Key Concepts

Sequence Terms

In mathematics, a sequence is an ordered list of numbers. Each number in a sequence is called a term. The terms of a sequence can be defined in various ways, including explicitly or recursively.
In the explicit method, a formula gives the nth term directly. For example, if the sequence is defined as \(a_n = 2n + 1\), you can find any term by plugging in the value of \(n\).
In contrast, recursive sequences define each term based on the previous ones. This means you need some initial terms and a specific rule that shows how to obtain any term from its predecessors. The problem in the given exercise defines a sequence recursively. The first term is \(a_1 = 3\), and subsequent terms are found using a recursive relationship.

Recursive Formula

A recursive formula specifies how to compute terms of a sequence based on its previous terms. In the exercise, the formula for calculating any term \(a_{k+1}\) is based on the preceding term and is given by\[a_{k+1} = 2(a_k - 1)\]
With recursive formulas, knowing the initial term is crucial as you use it to generate all subsequent terms.
Recursive formulas are quite powerful and can express sequences that grow or change in complex patterns. Each time the recursive formula is applied, it describes a step in the mathematical recursion that builds the sequence.

Mathematical Recursion

Mathematical recursion is a process where the solution to a problem depends on solutions to smaller instances of the same problem. It is a fundamental concept in defining recursive sequences.
When you use recursion in sequences, you repeatedly apply a formula or set of rules to previous terms to get new terms. This involves recursion steps and a base case:

• The base case provides a starting point, like \(a_1 = 3\) in our example.
• The recursive steps are the repeated application of the formula \(a_{k+1} = 2(a_k - 1)\).

Understanding recursion helps in recognizing patterns and predicting future values in sequences, making it a versatile tool in solving mathematical problems that involve sequences.