Problem 24 - Algebra 2

Question

Find the indicated term of each geometric sequence. $$ a_{1}=\frac{1}{3}, r=3, n=8 $$

Step-by-Step Solution

Answer

The 8th term is 729.

Step 1: Identify the Formula

To find the nth term of a geometric sequence, we use the formula: \( a_n = a_1 imes r^{(n-1)} \), where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number we want to find.

Step 2: Substitute Known Values

Substitute the given values into the formula: \( a_1 = \frac{1}{3} \), \( r = 3 \), and \( n = 8 \). The formula becomes: \( a_8 = \frac{1}{3} \times 3^{(8-1)} \).

Step 3: Simplify the Exponent Calculation

Calculate the exponent: \( 3^{7} \). Start by calculating step-by-step: \( 3^{2} = 9 \), \( 3^{3} = 27 \), and continue up to \( 3^{7} = 2187 \).

Step 4: Calculate the nth Term

Substitute \( 3^{7} = 2187 \) back into the expression \( a_8 = \frac{1}{3} \times 2187 \). Simplify this multiplication: \( a_8 = \frac{2187}{3} \).

Step 5: Simplify the Fraction

Divide \( 2187 \) by \( 3 \) to find the result of the fraction: \( 2187 \div 3 = 729 \). Thus, \( a_8 = 729 \).

Key Concepts

nth term formula

To find any term in a geometric sequence, a specific formula is used, called the nth term formula. This formula helps us calculate the term number we need.
The generic expression for finding the nth term is:

• \( a_n = a_1 \times r^{(n-1)} \)

Here,
\( a_1 \) is the first term of the sequence that we know.
\( r \), the common ratio, tells us how the sequence is changing.
\( n \) is the number of the term we want to calculate.
Use this formula by inserting the known values for these variables. This formula is essential to solve the problem of finding a particular term when dealing with geometric sequences.

common ratio

The common ratio, represented by \( r \), is crucial for understanding how a geometric sequence progresses.
In a geometric sequence, each term is obtained by multiplying the previous term by this common ratio.
This quality makes geometric sequences unique as each term grows or decreases at a consistent rate.
For example, if your sequence starts with the first term \( a_1 = \frac{1}{3} \) and \( r = 3 \), you would multiply \( \frac{1}{3} \) by \( 3 \) to get the next term.
As you progress down the sequence, you continue multiplying by \( r \) to find subsequent terms.
Understanding the common ratio doesn’t just help in finding new terms, but it also tells us about the general pattern of the sequence.

exponent calculation

Exponent calculation is a pivotal part of working with geometric sequences. In our formula \( a_n = a_1 \times r^{(n-1)} \), the exponent \( (n-1) \) is what tells you how many times you multiply the common ratio
by itself before multiplying by the first term.
Breaking down exponent calculation:

• \( r^{0} = 1 \)
• \( r^{1} = r \)
• \( r^{2} = r \times r \)

And so on. This pattern continues with every increase in powerWhen calculating \( r^{7} \) or any higher power, remember to carefully multiply each step. Exponent calculation ensures accurate scaling of terms in the sequence, crucial for obtaining the correct result.

fraction simplification

Simplifying fractions is the final step to reach the answer in our nth term problem.
Once you've calculated \( a_n = \frac{a \times r^{(n-1)}}{1} \), there might be fractions involved that need simplification.
Here's how it works:

• If you reach \( \frac{2187}{3} \), perform the division by dividing the numerator by the denominator.
• This gives \( 2187 \div 3 = 729 \). No fractions are left, giving a clean integer result.

Simplifying fractions ensures that we present our final result cleanly and understandably. When working with geometric sequences, especially those involving fractions, mastering this step is critical for an accurate solution.