Problem 47 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{1}=16,807, r=\frac{3}{7}, n=6 $$

Step-by-Step Solution

Answer

The 6th term is 243.

Step 1: Understanding the Problem

In this problem, we are given a geometric sequence with its first term \( a_1 = 16,807 \), the common ratio \( r = \frac{3}{7} \), and we need to find the 6th term of the sequence \( a_6 \).

Step 2: Using the Formula for nth Term

The formula for the \( n \)-th term of a geometric sequence is given by \( a_n = a_1 \, r^{n-1} \). Since we want the 6th term, we will use \( n = 6 \).

Step 3: Substituting the Values

Substitute the given values into the formula: \[a_6 = 16,807 \cdot \left(\frac{3}{7}\right)^{6-1}\]This simplifies to: \[a_6 = 16,807 \cdot \left(\frac{3}{7}\right)^{5}\]

Step 4: Calculating the Power of the Ratio

Calculate \( \left(\frac{3}{7}\right)^5 \). First, calculate \( \frac{3}{7} \times \frac{3}{7} = \frac{9}{49} \), \( \frac{9}{49} \times \frac{3}{7} = \frac{27}{343} \), \( \frac{27}{343} \times \frac{3}{7} = \frac{81}{2401} \), and finally, \( \frac{81}{2401} \times \frac{3}{7} = \frac{243}{16807} \).

Step 5: Find the 6th term

Now substitute \( \left(\frac{3}{7}\right)^5 = \frac{243}{16807} \) back into the equation: \[a_6 = 16,807 \cdot \frac{243}{16807}\]Simplifying, we find:\[a_6 = 243\]

Step 6: Conclusion

The 6th term \( a_6 \) in the geometric sequence is 243.

Key Concepts

Understanding the nth Term Formula

In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio. To locate any specific term within a geometric sequence, we use the nth term formula. This formula is presented as:\[ a_n = a_1 \cdot r^{(n-1)} \]Here, \( a_n \) is the term number we seek, \( a_1 \) is the first term of the sequence, \( r \) is the common ratio, and \( n \) signifies the position of the term in the sequence.
By applying this formula, we can find any term in the sequence with ease as long as we know the first term, the common ratio, and the position of the desired term.
This method provides a reliable way to understand and calculate terms in a geometric sequence.

The Role of Common Ratio

The common ratio in a geometric sequence determines how each term relates to its predecessor. It is a constant factor by which each term is multiplied to get the next term in the sequence.
This ratio is a key component in calculating subsequent terms using the nth term formula.
For example, in a sequence where the common ratio \( r = \frac{3}{7} \), every term is represented by multiplying the previous term by \( \frac{3}{7} \). It's crucial to understand that the common ratio can be smaller than 1, resulting in terms that gradually decrease, or larger than 1, leading to terms that increase.
Understanding and identifying this ratio is crucial as it establishes the nature and behavior of the sequence throughout its generation.

Mastering Exponentiation

Exponentiation involves raising a base number to the power of an exponent, representing repeated multiplication of the base.In a geometric sequence, exponentiation is vital because the common ratio is raised to the power of \( (n-1) \) in the nth term formula.
For instance, in our given sequence example, we need to compute \( \left(\frac{3}{7}\right)^{5} \) which simplifies to \( \frac{243}{16807} \). This involves continuously multiplying \( \frac{3}{7} \) by itself five times to find the exact value.
The process involves thorough understanding and careful calculation to ensure accurate outcomes. Therefore, becoming comfortable with this mathematical operation can significantly enhance your ability to work with geometric sequences.

Sequence Calculation Simplified

Sequence calculation in geometric sequences is a straightforward application of the nth term formula once you understand the components involved.To find any term, we:

• Identify the first term \( a_1 \)
• Recognize the common ratio \( r \)
• Use the nth term position \( n \)

In our exercise, by plugging values into the formula and performing step-by-step computations, we arrive at the desired sequence term.
Recall that each component plays a crucial role: the first term establishes the starting point, the common ratio determines the factor of progression between terms, and exponentiation orders the multiplication process.
With these steps, you can confidently navigate sequence calculations and determine terms effectively.