Problem 16 - Algebra and Trigonometry

Question

are defined using recursion formulas. Write the first four terms of each sequence. $$ a_{1}=2 \text { and } a_{n}=5 a_{n-1} \text { for } n \geq 2 $$

Step-by-Step Solution

Answer

The first four terms of the sequence are: \(a_1 = 2\), \(a_2 = 10\), \(a_3 = 50\), \(a_4 = 250\)

Step 1: Compute the second term

Calculate the value of \(a_2\) by multiplying the first term (\(a_1=2\)) by 5. That will be \(a_{2}= 5 \cdot a_{1} = 10\)

Step 2: Compute the third term

Calculation of \(a_3\) is done by multiplying the second term (\(a_2=10\)) by 5. Therefore, \(a_{3}= 5 \cdot a_{2} = 50\)

Step 3: Compute the fourth term

Calculate the value of \(a_4\) by multiplying the third term (\(a_3=50\)) by 5. That gives \(a_{4}= 5 \cdot a_{3} = 250\)

Key Concepts

Recursion Formula

A recursion formula is a way of defining the terms of a sequence using the previous terms. In the given exercise, each term of the sequence is found by multiplying the previous term by 5. This is noted in the recursive formula: \(a_n = 5a_{n-1}\) for \(n \geq 2\).
This formula explains how to derive the term \(n\) from the term \(n-1\), making the pattern of the sequence quite predictable.
Recursion formulas are useful in various fields, including mathematics and computer science, as they allow for the generation of complex sequences from simple rules.

• Start with an initial term, such as \(a_1 = 2\).
• Apply the recursion formula to find subsequent terms.
• In our example, the second term becomes \(a_2 = 5 \times a_1\).
• This step-by-step approach continues until you reach the desired number of terms.

Consistently using a recursion formula makes calculating each term systematic and straightforward.

Geometric Sequence

A geometric sequence is a type of sequence where each term is derived by multiplying the previous term by a fixed, non-zero number, known as the common ratio. In our exercise, the sequence is geometric because each term is determined by multiplying the previous term by 5.
The first term is given as \(a_1 = 2\) and each subsequent term is obtained by applying the common ratio, which is 5 in this case.

• The first term, \(a_1 = 2\), sets the foundation.
• The second term, \(a_2 = 5 \times 2 = 10\), shows how the common ratio works.
• The pattern continues, identifying this as a characteristic geometric sequence.

Geometric sequences are easy to identify once you recognize the repeated multiplication pattern.

Sequence Terms

Sequence terms represent the individual elements within a sequence. In our exercise, you determined the sequence terms using a recursive process.
The process started with an initial term, and each term was generated using a multiplication operation. For example, beginning with \(a_1 = 2\), the second term was found as \(a_2 = 10\).

• \(a_1\) is the base term, often provided in the problem statement.
• \(a_2\), \(a_3\), etc., are computed using the recursive formula: \(a_n = 5a_{n-1}\).
• The sequence exhibits exponential growth due to the consistent multiplication.

Understanding sequence terms and how they grow helps in identifying patterns and predicting future terms.