If 40, n, and 10 form the first three terms of a geometric sequence, then which of the following is the value of n? (1) 16 (3) 20 (2) 18 (4) 24

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Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Geometric Sequence Problem**

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**Question 15:**

If 40, \( n \), and 10 form the first three terms of a geometric sequence, then which of the following is the value of \( n \)?

1. 16
2. 18
3. 20
4. 24

**Answer Choices:**
- ( ) 1
- ( ) 2
- ( ) 3
- ( ) 4

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Explanation:
In a geometric sequence, each term after the first is the product of the previous term and a constant ratio \( r \). That is, the sequence can be represented as:
\[ a, ar, ar^2, ... \]

Given that 40, \( n \), and 10 form the first three terms of a geometric sequence:
- The first term \( a \) is 40.
- The second term \( n \) is \( ar \).
- The third term is 10.

Since the third term is \( ar^2 \), we have:
\[ ar^2 = 10 \]

From the first term:
\[ a = 40 \]

The second term:
\[ ar = n \]
\[ 40r = n \]

Substituting \( a = 40 \) into the equation \( ar^2 = 10 \):
\[ 40r^2 = 10 \]
\[ r^2 = \frac{10}{40} \]
\[ r^2 = \frac{1}{4} \]
\[ r = \frac{1}{2} \, \text{or} \, r = -\frac{1}{2} \]

If  \( r = \frac{1}{2} \):
\[ n = 40 \times \frac{1}{2} = 20 \]

Therefore, \( n = 20 \), which corresponds to option 3.

Correct Answer: **3. 20**

Remember that identifying sequences and understanding their properties are crucial skills in mathematics that apply widely, from algebra to advanced calculus.
Transcribed Image Text:**Geometric Sequence Problem** --- **Question 15:** If 40, \( n \), and 10 form the first three terms of a geometric sequence, then which of the following is the value of \( n \)? 1. 16 2. 18 3. 20 4. 24 **Answer Choices:** - ( ) 1 - ( ) 2 - ( ) 3 - ( ) 4 --- Explanation: In a geometric sequence, each term after the first is the product of the previous term and a constant ratio \( r \). That is, the sequence can be represented as: \[ a, ar, ar^2, ... \] Given that 40, \( n \), and 10 form the first three terms of a geometric sequence: - The first term \( a \) is 40. - The second term \( n \) is \( ar \). - The third term is 10. Since the third term is \( ar^2 \), we have: \[ ar^2 = 10 \] From the first term: \[ a = 40 \] The second term: \[ ar = n \] \[ 40r = n \] Substituting \( a = 40 \) into the equation \( ar^2 = 10 \): \[ 40r^2 = 10 \] \[ r^2 = \frac{10}{40} \] \[ r^2 = \frac{1}{4} \] \[ r = \frac{1}{2} \, \text{or} \, r = -\frac{1}{2} \] If \( r = \frac{1}{2} \): \[ n = 40 \times \frac{1}{2} = 20 \] Therefore, \( n = 20 \), which corresponds to option 3. Correct Answer: **3. 20** Remember that identifying sequences and understanding their properties are crucial skills in mathematics that apply widely, from algebra to advanced calculus.
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