How to Find the Sum of a Geometric Sequence - Owlcation - Free Printable

The image shows three sequences of numbers. Let's analyze each sequence step by step to identify the pattern and determine the next terms.

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Sequence 1: \( 4, 8, 16, \ldots \)

#### Step 1: Identify the pattern
- The first term is \( 4 \).
- The second term is \( 8 \), which is \( 4 \times 2 \).
- The third term is \( 16 \), which is \( 8 \times 2 \).

This suggests that each term is obtained by multiplying the previous term by \( 2 \).

#### Step 2: Generalize the pattern
The sequence is a geometric progression where the first term \( a = 4 \) and the common ratio \( r = 2 \). The \( n \)-th term of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]
For this sequence:
\[
a_n = 4 \cdot 2^{n-1}
\]

#### Step 3: Find the next term
The next term (fourth term) is:
\[
a_4 = 4 \cdot 2^{4-1} = 4 \cdot 2^3 = 4 \cdot 8 = 32
\]

So, the sequence continues as:
\[
4, 8, 16, 32, \ldots
\]

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Sequence 2: \( 4, 12, 36, \ldots \)

#### Step 1: Identify the pattern
- The first term is \( 4 \).
- The second term is \( 12 \), which is \( 4 \times 3 \).
- The third term is \( 36 \), which is \( 12 \times 3 \).

This suggests that each term is obtained by multiplying the previous term by \( 3 \).

#### Step 2: Generalize the pattern
The sequence is a geometric progression where the first term \( a = 4 \) and the common ratio \( r = 3 \). The \( n \)-th term of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]
For this sequence:
\[
a_n = 4 \cdot 3^{n-1}
\]

#### Step 3: Find the next term
The next term (fourth term) is:
\[
a_4 = 4 \cdot 3^{4-1} = 4 \cdot 3^3 = 4 \cdot 27 = 108
\]

So, the sequence continues as:
\[
4, 12, 36, 108, \ldots
\]

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Sequence 3: \( 52, 26, 13, 6.5, \ldots \)

#### Step 1: Identify the pattern
- The first term is \( 52 \).
- The second term is \( 26 \), which is \( 52 \div 2 \).
- The third term is \( 13 \), which is \( 26 \div 2 \).
- The fourth term is \( 6.5 \), which is \( 13 \div 2 \).

This suggests that each term is obtained by dividing the previous term by \( 2 \).

#### Step 2: Generalize the pattern
The sequence is a geometric progression where the first term \( a = 52 \) and the common ratio \( r = \frac{1}{2} \). The \( n \)-th term of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]
For this sequence:
\[
a_n = 52 \cdot \left(\frac{1}{2}\right)^{n-1}
\]

#### Step 3: Find the next term
The next term (fifth term) is:
\[
a_5 = 52 \cdot \left(\frac{1}{2}\right)^{5-1} = 52 \cdot \left(\frac{1}{2}\right)^4 = 52 \cdot \frac{1}{16} = \frac{52}{16} = 3.25
\]

So, the sequence continues as:
\[
52, 26, 13, 6.5, 3.25, \ldots
\]

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Final Answer

Combining the results from all three sequences, we have:
1. \( 4, 8, 16, 32, \ldots \)
2. \( 4, 12, 36, 108, \ldots \)
3. \( 52, 26, 13, 6.5, 3.25, \ldots \)

Thus, the next terms are:
\[
\boxed{32, 108, 3.25}
\]