64 tuning forks are arranged in order of increasing frequency and any two successive forks give foor beats per second when sounded together. If the last fork gives the octave of the first, calculate the frequency of the latter.

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the frequency of the first tuning fork
Let the frequency of the first tuning fork be \( n \) Hz.

### Step 2: Define the frequency of the second tuning fork
Since any two successive tuning forks give 4 beats per second, the frequency of the second tuning fork can be expressed as:
\[ \text{Frequency of 2nd fork} = n + 4 \text{ Hz} \]

### Step 3: Generalize the frequency of the x-th tuning fork
For the x-th tuning fork, the frequency can be expressed as:
\[ \text{Frequency of x-th fork} = n + 4(x - 1) \text{ Hz} \]

### Step 4: Define the frequency of the 64th tuning fork
For the 64th tuning fork, we can write:
\[ \text{Frequency of 64th fork} = n + 4(64 - 1) = n + 4 \times 63 = n + 252 \text{ Hz} \]

### Step 5: Use the given information about the octave
According to the problem, the frequency of the last fork (64th) is the octave of the first fork. The octave means that the frequency of the 64th fork is double that of the first fork:
\[ n + 252 = 2n \]

### Step 6: Solve for n
Rearranging the equation gives:
\[ 252 = 2n - n \]
\[ 252 = n \]

### Step 7: Calculate the frequency of the first tuning fork
Thus, the frequency of the first tuning fork is:
\[ n = 252 \text{ Hz} \]

### Step 8: Calculate the frequency of the 64th tuning fork
Now, we can find the frequency of the 64th tuning fork:
\[ \text{Frequency of 64th fork} = 2n = 2 \times 252 = 504 \text{ Hz} \]

### Final Answer
The frequency of the first tuning fork is **252 Hz** and the frequency of the 64th tuning fork is **504 Hz**.

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