32 tuning forks are arranged in order of increasing frequency such that frequency of last fork is double than first. 6 beats per second are produced by each two consecutive forks. The frequency of last fork is `n xx 186`. Find the value of n.
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Explanation

To solve the problem, we need to analyze the information given about the tuning forks and their frequencies.

### Step-by-Step Solution:

1. **Understanding the Arrangement of Tuning Forks**:
- We have 32 tuning forks arranged in order of increasing frequency.
- Let the frequency of the first tuning fork be denoted as \( f \).

2. **Frequency of the Last Tuning Fork**:
- According to the problem, the frequency of the last tuning fork is double that of the first. Therefore, the frequency of the last tuning fork can be expressed as:
\[ f_{last} = 2f \]

3. **Beats Produced by Consecutive Forks**:
- The problem states that 6 beats per second are produced by each pair of consecutive forks. This implies that the difference in frequency between any two consecutive forks is 6 Hz.
- Thus, if the frequency of the first fork is \( f \), the frequency of the second fork will be:
\[ f_2 = f + 6 \]
- Continuing this pattern, we can express the frequency of the \( n \)-th fork as:
\[ f_n = f + (n-1) \times 6 \]
- For the 32nd fork (last fork), we have:
\[ f_{32} = f + 31 \times 6 \]

4. **Setting Up the Equation**:
- We know from the earlier step that \( f_{last} = 2f \). Therefore, we can set up the equation:
\[ f + 31 \times 6 = 2f \]

5. **Solving for \( f \)**:
- Rearranging the equation gives us:
\[ 31 \times 6 = 2f - f \]
\[ 186 = f \]
- Thus, the frequency of the first tuning fork is \( f = 186 \) Hz.

6. **Finding the Frequency of the Last Fork**:
- Now, substituting \( f \) back into the equation for the last fork:
\[ f_{last} = 2f = 2 \times 186 = 372 \text{ Hz} \]

7. **Expressing the Last Fork's Frequency**:
- The problem states that the frequency of the last fork is given as \( n \times 186 \). Therefore, we can write:
\[ n \times 186 = 372 \]
- Dividing both sides by 186 gives:
\[ n = \frac{372}{186} = 2 \]

### Final Answer:
The value of \( n \) is \( 2 \).

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