Chapter8: Sequences, Series,and Probability
Section8.3: Geometric Sequences And Series
Problem 1ECP: Write the first four terms of the geometric sequence whose nth term is 6(2)n. Then find the common...
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![## Finding the First Four Terms of a Given Sequence
**Problem Statement:**
Find the first four terms of the sequence given by the following formula:
\[ a_n = \frac{3^n + 5}{2^n}, \quad n = 1, 2, 3, \ldots \]
**Steps to Solve:**
1. **Substitute \(n = 1\)** into the sequence formula:
\[
a_1 = \frac{3^1 + 5}{2^1} = \frac{3 + 5}{2} = \frac{8}{2} = 4
\]
2. **Substitute \(n = 2\)** into the sequence formula:
\[
a_2 = \frac{3^2 + 5}{2^2} = \frac{9 + 5}{4} = \frac{14}{4} = 3.5
\]
3. **Substitute \(n = 3\)** into the sequence formula:
\[
a_3 = \frac{3^3 + 5}{2^3} = \frac{27 + 5}{8} = \frac{32}{8} = 4
\]
4. **Substitute \(n = 4\)** into the sequence formula:
\[
a_4 = \frac{3^4 + 5}{2^4} = \frac{81 + 5}{16} = \frac{86}{16} = 5.375
\]
**First Four Terms of the Sequence:**
Thus, the first four terms of the given sequence are:
\[ 4, 3.5, 4, 5.375 \]
**Graphical Representation:**
There are no specific graphs or diagrams provided in this problem.
Note: The empty boxes following the problem statement are meant for the student to fill in their answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14a97047-56a8-420b-97d8-c66168d5d651%2F99581052-7112-4301-8419-1563c4ce8396%2Fkdn1w8h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:## Finding the First Four Terms of a Given Sequence
**Problem Statement:**
Find the first four terms of the sequence given by the following formula:
\[ a_n = \frac{3^n + 5}{2^n}, \quad n = 1, 2, 3, \ldots \]
**Steps to Solve:**
1. **Substitute \(n = 1\)** into the sequence formula:
\[
a_1 = \frac{3^1 + 5}{2^1} = \frac{3 + 5}{2} = \frac{8}{2} = 4
\]
2. **Substitute \(n = 2\)** into the sequence formula:
\[
a_2 = \frac{3^2 + 5}{2^2} = \frac{9 + 5}{4} = \frac{14}{4} = 3.5
\]
3. **Substitute \(n = 3\)** into the sequence formula:
\[
a_3 = \frac{3^3 + 5}{2^3} = \frac{27 + 5}{8} = \frac{32}{8} = 4
\]
4. **Substitute \(n = 4\)** into the sequence formula:
\[
a_4 = \frac{3^4 + 5}{2^4} = \frac{81 + 5}{16} = \frac{86}{16} = 5.375
\]
**First Four Terms of the Sequence:**
Thus, the first four terms of the given sequence are:
\[ 4, 3.5, 4, 5.375 \]
**Graphical Representation:**
There are no specific graphs or diagrams provided in this problem.
Note: The empty boxes following the problem statement are meant for the student to fill in their answers.
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