Problem 35 - Algebra and Trigonometry

Question

The common ratio in a geometric sequence is \(\frac{2}{5},\) and the fourth term is \(\frac{5}{2} .\) Find the third term.

Step-by-Step Solution

Answer

The third term is \( \frac{25}{4} \).

Step 1: Understanding the Geometric Sequence Formula

A geometric sequence is defined by the formula for the nth term: \( a_n = a_1 imes r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number. Our objective is to find the third term.

Step 2: Establish the Known Information

We know that the fourth term \( a_4 = \frac{5}{2} \) and the common ratio \( r = \frac{2}{5} \). We will use these known values to find the third term.

Step 3: Apply the Fourth Term Formula

Using \( a_4 = a_1 imes r^{3} = \frac{5}{2} \), substitute for \( r \) to get: \( a_1 \times \left( \frac{2}{5} \right)^3 = \frac{5}{2} \).

Step 4: Simplify and Solve for \( a_1 \)

Simplify \( \left( \frac{2}{5} \right)^3 = \frac{8}{125} \), so the equation becomes: \( a_1 \times \frac{8}{125} = \frac{5}{2} \). Multiply both sides by \( \frac{125}{8} \) to solve for \( a_1 \). We find that \( a_1 = \frac{5}{2} \times \frac{125}{8} \).

Step 5: Calculate \( a_1 \)

Calculate \( a_1 = \frac{5 \times 125}{2 \times 8} = \frac{625}{16} \). Thus, the first term \( a_1 = \frac{625}{16} \).

Step 6: Find the Third Term

Using the formula for the third term \( a_3 = a_1 \times r^{2} \), substitute \( a_1 = \frac{625}{16} \) and \( r = \frac{2}{5} \). So, \( a_3 = \frac{625}{16} \times \left( \frac{2}{5} \right)^2 = \frac{625}{16} \times \frac{4}{25} \).

Step 7: Simplify the Third Term

Calculate \( a_3 = \frac{625 \times 4}{16 \times 25} = \frac{2500}{400} = \frac{250}{40} = \frac{25}{4} \). Therefore, the third term is \( \frac{25}{4} \).

Key Concepts

Common Ratio

In a geometric sequence, the common ratio is a key concept. It is the constant factor that you multiply by to get from one term to the next. For example, in the problem provided, the common ratio is \( \frac{2}{5} \). This means that each term in the sequence is \( \frac{2}{5} \) times the previous term.
The common ratio can be any non-zero number, and it determines the direction and rate of growth or decay of the sequence:

• If \( r > 1 \), the sequence grows exponentially.
• If \( 0 < r < 1 \), the sequence decreases and approaches zero but never reaches it.
• If \( r = 1 \), all terms are equal to the first term.
• If \( r < 0 \), terms will alternate in sign.

Understanding the common ratio helps you comprehend how each term fits into the overall structure of the sequence.

Nth Term Formula

The nth term formula for a geometric sequence is a fundamental tool to find any term in the sequence. It is given by \( a_n = a_1 \times r^{(n-1)} \), where:

• \( a_n \) is the nth term you're trying to find.
• \( a_1 \) is the first term of the sequence.
• \( r \) is the common ratio.
• \( n \) is the position of the term in the sequence.

Knowing this formula allows you to calculate any term if you have the first term and the common ratio. The power \( r^{(n-1)} \) tells you how many times you multiply the first term by the common ratio.
For example, using the nth term formula, you can find the third term (\( a_3 \)) or the fourth term (\( a_4 \)) once you have \( a_1 \) and \( r \) handy.

Fourth Term

To find the fourth term in a geometric sequence, you use the nth term formula, substituting \( n = 4 \). For instance, in our scenario, the fourth term \( a_4 \) is given as \( \frac{5}{2} \).
This information is crucial because it acts as a checkpoint, allowing us to reverse-engineer the sequence and find other terms.
By rewriting the formula to express \( a_1 \) in terms of the given fourth term, we acknowledge:

• \( a_4 = a_1 \times r^3 \)

This relationship lets you derive \( a_1 \), which is the cornerstone for calculating any other terms like the third term.

Third Term

The task of finding the third term of a geometric sequence is an interesting exercise in applying the nth term formula. In our example, once we know the first term \( a_1 \) and the common ratio \( r \), we can use the formula \( a_3 = a_1 \times r^2 \) to find the third term.
Start by substituting the values you have:

• \( a_1 = \frac{625}{16} \)
• \( r = \frac{2}{5} \)

This means:

• \( a_3 = \frac{625}{16} \times \left( \frac{2}{5} \right)^2 \)

Simplify the expression to get \( a_3 = \frac{25}{4} \).
This step-by-step approach reinforces your understanding of geometric sequences and highlights the importance of using known values correctly to find unknown terms.