Problem 39 - Algebra and Trigonometry

Question

A sequence is defined recursively. List the first five terms. \(a_{1}=4 ; \quad a_{n}=3 a_{n-1}\)

Step-by-Step Solution

Answer

The first five terms of the sequence are 4, 12, 36, 108, and 324.

Step 1: Understanding the Initial Term

Identify the initial term of the sequence, which is given by the problem. Here, it is provided that \(a_{1} = 4\).

Step 2: Determining the Second Term

Use the recursive formula \(a_{n} = 3a_{n-1}\) to calculate the second term. Since \(a_{2} = 3a_{1}\), we have \(a_{2} = 3 \times 4 = 12\).

Step 3: Calculating the Third Term

Using the same recursive formula, find the third term: \(a_{3} = 3a_{2}\). Thus, \(a_{3} = 3 \times 12 = 36\).

Step 4: Finding the Fourth Term

Apply the formula to determine the fourth term: \(a_{4} = 3a_{3}\). Therefore, \(a_{4} = 3 \times 36 = 108\).

Step 5: Computing the Fifth Term

Finally, use the recursive relationship to find the fifth term: \(a_{5} = 3a_{4}\). This gives \(a_{5} = 3 \times 108 = 324\).

Key Concepts

initial term

In any sequence defined recursively, the initial term is the starting point. It serves as the first value in the sequence and is crucial because all subsequent terms are derived from it. For the sequence in our exercise, the initial term is given as \(a_{1} = 4\). Understanding the initial term is important because it plugs directly into the recursive formula to start generating additional terms.

recursive formula

A recursive formula defines each term in a sequence using the preceding term(s). In the given exercise, the recursive formula is \(a_{n} = 3a_{n-1}\). This means that to find any term \(a_{n}\), you take the previous term \(a_{n-1}\) and multiply it by 3. Recursive formulas are powerful because they provide a straightforward way to calculate a sequence without needing an explicit formula for \(a_{n}\). Understanding how to apply this formula step-by-step lets you build the entire sequence from the initial term.

sequence calculation

Calculating the terms of a sequence using a recursive formula involves a series of steps. Here’s how we can do it for our exercise:

• **First Term**: Given directly as \(a_{1} = 4 \).
• **Second Term**: Use the recursive formula, \(a_{2} = 3a_{1} = 3 \times 4 = 12 \).
• **Third Term**: Again using the formula, \(a_{3} = 3a_{2} = 3 \times 12 = 36 \).
• **Fourth Term**: Apply the formula, \(a_{4} = 3a_{3} = 3 \times 36 = 108 \).
• **Fifth Term**: Finally, \(a_{5} = 3a_{4} = 3 \times 108 = 324 \).

This step-by-step approach ensures you correctly apply the recursive formula to build out the entire sequence.

geometric progression

The sequence in the exercise is also a geometric progression. A geometric progression is a sequence where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In our case:

• **Common Ratio**: The fixed number multiplying each term to get the next is 3.
• **Initial Term**: Starting from 4.
• **Formula**: \(a_{n} = 4 \times 3^{n-1} \), since \(a_{1} = 4 \) and the ratio is 3.

Understanding that this sequence is a geometric progression helps recognize patterns and derive properties related to the sequence, such as sums and specific term calculations.