Problem 29 - College Algebra

Question

For the following exercises, write a recursive formula for each geometric sequence. $$ a_{n}=\left\\{\frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360}, \ldots\right\\} $$

Step-by-Step Solution

Answer

Recursive formula: \( a_1 = \frac{3}{5}, a_n = a_{n-1} \times \frac{1}{6} \) for \( n > 1 \).

Step 1: Identify the first term

The first term of the sequence, denoted as \( a_1 \), is the initial value in the sequence. For the given sequence \( \left\{\frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360}, \ldots\right\} \), the first term is \( a_1 = \frac{3}{5} \).

Step 2: Determine the common ratio

In a geometric sequence, the common ratio \( r \) is found by dividing any term by the previous term. Divide the second term by the first term: \( \frac{1}{10} \div \frac{3}{5} = \frac{1}{10} \times \frac{5}{3} = \frac{1}{6} \). Verify this by dividing the third term by the second term: \( \frac{1}{60} \div \frac{1}{10} = \frac{1}{6} \). The common ratio is \( r = \frac{1}{6} \).

Step 3: Write the recursive formula

For a geometric sequence, the recursive formula expresses each term based on the previous term. By using the first term \( a_1 \) and the common ratio \( r \), the recursive formula is: \[ a_n = a_{n-1} \times \frac{1}{6} \text{ for } n > 1 \] The first term is specified as \( a_1 = \frac{3}{5} \). The complete recursive description is: \[ a_1 = \frac{3}{5}, \quad a_n = a_{n-1} \times \frac{1}{6} \text{ for } n > 1 \]

Key Concepts

Recursive Formula in Geometric Sequences

A recursive formula in a geometric sequence allows you to find any term based on its previous term. This is a great tool for sequences where each term is consistent with a particular multiplication pattern. In the given sequence: \( \left\{\frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360}, \ldots\right\} \), we use a recursive formula to systematically express the relationship between terms:

• We start with the first term, \( a_1 \), which will form the basis of our equation.
• Each subsequent term can be calculated by multiplying the preceding term by the common ratio \( r \).
• In this scenario, our recursive relationship takes the form \( a_n = a_{n-1} \times \frac{1}{6} \) for \( n > 1 \).

This means that every term after the first is derived by multiplying the preceding term by \( \frac{1}{6} \). The use of recursive formulas is efficient because we don't always need to know every prior term to find a new one, just the last one calculated.

Identifying the First Term

The first term, represented symbolically as \( a_1 \), serves as the starting point of a geometric sequence. Knowing the initial term is essential because it grounds our recursive formula, giving us a concrete beginning from which all subsequent elements develop. In our example sequence \( a_n = \left\{\frac{3}{5}, \frac{1}{10}, \frac{1}{60}, \frac{1}{360}, \ldots\right\} \):

• The expression \( a_1 = \frac{3}{5} \) defines our first term.
• This initial term is crucial in applying the recursive formula, as each following term is a multiple of the previous one and the common ratio.
• Having a clear first term is vital, ensuring that the sequence proceeds systematically.

Remember, the first term doesn't change within a geometric sequence. It's the baseline from which progression follows, and without it, the structure of the sequence wouldn't hold.

Understanding the Common Ratio

The common ratio \( r \) in a geometric sequence defines how each term relates to the one before it. It's obtained by dividing any term by the one preceding it and remains constant throughout the sequence. Let's look at the sequence at hand:

• Starting from the second term \( \frac{1}{10} \) and dividing it by the first term \( \frac{3}{5} \) gives \( \frac{1}{6} \).
• The consistency of this ratio is confirmed when \( \frac{1}{60} \) divided by \( \frac{1}{10} \) also results in \( \frac{1}{6} \).
• Thus, our common ratio is derived as \( r = \frac{1}{6} \).

This ratio is integral to the recursive formula, \( a_n = a_{n-1} \times r \), providing the consistent multiplier by which each term evolves from its predecessor. Understanding this concept allows you to see the uniform growth or shrinkage across the sequence.