There are 26 tuning forks arranged in the decreasing order of their frequencies. Each tuning fork gives 3 beats with the next. The first one is octave of the last. What is the frequency of 18th tuning fork ?

Options

A. 100 Hz
B. 99 Hz
C. 96 Hz
D. 103 Hz

Explanation

To find the frequency of the 18th tuning fork, we can follow these steps:

### Step 1: Understand the relationship between the tuning forks
We know that there are 26 tuning forks arranged in decreasing order of frequency. Each tuning fork produces 3 beats with the next one, which means the difference in frequency between each consecutive tuning fork is 3 Hz.

### Step 2: Define the frequencies
Let the frequency of the last tuning fork (the 26th fork) be \( n \). Since the first tuning fork (the 1st fork) is an octave of the last one, its frequency will be \( 2n \).

### Step 3: Write the frequency of each tuning fork
The frequencies of the tuning forks can be expressed as follows:
- 1st tuning fork: \( 2n \)
- 2nd tuning fork: \( 2n - 3 \)
- 3rd tuning fork: \( 2n - 6 \)
- ...
- 26th tuning fork: \( n \)

The general formula for the frequency of the \( k \)-th tuning fork can be written as:
\[ f_k = 2n - 3(k-1) \]

### Step 4: Find the frequency of the 26th tuning fork
Using the formula for the 26th tuning fork:
\[ f_{26} = 2n - 3(26-1) = 2n - 75 \]
Since \( f_{26} = n \), we can set up the equation:
\[ n = 2n - 75 \]

### Step 5: Solve for \( n \)
Rearranging the equation gives:
\[ 75 = 2n - n \]
\[ n = 75 \text{ Hz} \]

### Step 6: Find the frequency of the 18th tuning fork
Now we can find the frequency of the 18th tuning fork using the formula:
\[ f_{18} = 2n - 3(18-1) = 2n - 51 \]
Substituting \( n = 75 \):
\[ f_{18} = 2(75) - 51 = 150 - 51 = 99 \text{ Hz} \]

### Final Answer
The frequency of the 18th tuning fork is **99 Hz**.

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