Question

Find the indicated term of each geometric sequence. 7th term of \(1, \frac{1}{2}, \frac{1}{4}\)

Short answer

The 7th term is \( \frac{1}{64} \).

Step-by-step solution

Identify the first term

The first term of the geometric sequence is given as 1.

Determine the common ratio

To find the common ratio, divide the second term by the first term: \[ r = \frac{\frac{1}{2}}{1} = \frac{1}{2} \].

Use the formula for the nth term

The formula for the nth term of a geometric sequence is given by \[ a_n = a_1 \cdot r^{(n-1)} \], where \( a_1 \) is the first term and \( r \) is the common ratio.

Substitute the given values

Substitute \( a_1 = 1 \), \( r = \frac{1}{2} \), and \( n = 7 \) into the formula: \[ a_7 = 1 \cdot \left( \frac{1}{2} \right)^{6} \].

Calculate the 7th term

Calculate \( \left( \frac{1}{2} \right)^{6} \): \[ \left( \frac{1}{2} \right)^{6} = \frac{1}{64} \]. Therefore, \[ a_7 = \frac{1}{64} \].

Key concepts

common ratio

In a geometric sequence, the 'common ratio' is a constant factor between consecutive terms. Identifying the common ratio is crucial for understanding how the sequence progresses.
For example, in the sequence given (1, \( \frac{1}{2} \), \( \frac{1}{4} \)), you can find the common ratio by dividing the second term by the first term. This is done as follows:
\[ r = \frac{\frac{1}{2}}{1} = \frac{1}{2} \]

Likewise, you can verify the ratio between other terms. For the third term, you divide \( \frac{1}{4} \) by \( \frac{1}{2} \), which will also give you \( \frac{1}{2} \). This confirms that the common ratio is \( \frac{1}{2} \).

Remembering that the same ratio applies to the entire sequence helps simplify calculations for terms far down the list.

nth term formula

The 'nth term formula' for a geometric sequence allows you to find any term in the sequence quickly and accurately. The formula is

\[ a_n = a_1 \cdot r^{(n-1)} \]

Here, \( a_n \) represents the 'nth term', \( a_1 \) is the 'first term', \( r \) is the 'common ratio', and \( n \) is the position of the term in the sequence.

For our example, the sequence starts with \( a_1 = 1 \) and has a common ratio \( r = \frac{1}{2} \). To find the 7th term (\( a_7 \)), you substitute the given values into the formula:
\[ a_7 = 1 \cdot \left( \frac{1}{2} \right)^{6} \]

This step simplifies the problem and makes calculations straightforward, ensuring you can quickly derive any term in the sequence you need.

sequence calculation

Performing 'sequence calculation' correctly requires applying the appropriate formulas and understanding the properties of geometric sequences.
To calculate the 7th term of the series 1, \( \frac{1}{2} \), \( \frac{1}{4} \), you first use the nth term formula:

\[ a_7 = 1 \cdot \left( \frac{1}{2} \right)^{6} \]

Next, break it down step by step. First, calculate \( \left( \frac{1}{2} \right)^6 \):

\[ \left( \frac{1}{2} \right)^{6} = \frac{1}{64} \]

So the 7th term is:

\[ a_7 = 1 \cdot \frac{1}{64} = \frac{1}{64} \]

This logical progression ensures accuracy and helps you clearly see how each term is derived.

algebra

Algebra plays a fundamental role in understanding and solving geometric sequences. Utilizing algebraic principles makes the processes of discovering terms and solving equations straightforward.

For geometric sequences specifically, you often rely on the power of exponents and properties of equality.

Consider another look at our nth term formula:

\[ a_n = a_1 \cdot r^{(n-1)} \]

This formula itself is purely algebraic, where the first term (\( a_1 \)), the common ratio (\( r \)), and the term number (\( n \)) all represent variables and constants that you manipulate algebraically. The exponential expression \( r^{(n-1)} \) showcases how exponents work in algebra.

By mastering these algebraic manipulations, you gain a strong foundation for navigating more complex mathematical topics with confidence.