For a certain stretched string, three consecutive resonance frequencies are observed as 105, 175 and 245 Hz respectively. Then, the fundamental frequency is

Answer

To find the fundamental frequency of the stretched string given the consecutive resonance frequencies, we can follow these steps:

### Step 1: Identify the given frequencies
The three consecutive resonance frequencies are:
- \( f_1 = 105 \, \text{Hz} \)
- \( f_2 = 175 \, \text{Hz} \)
- \( f_3 = 245 \, \text{Hz} \)

### Step 2: Understand the relationship between the frequencies
For a string fixed at one end, the resonance frequencies correspond to odd harmonics. The relationship can be expressed as:
- \( f_1 : f_2 : f_3 = 3 : 5 : 7 \)

### Step 3: Use the relationship to find the fundamental frequency
The fundamental frequency \( f_0 \) can be calculated using the formula:
- \( f_0 = \frac{f_1}{3} = \frac{f_2}{5} = \frac{f_3}{7} \)

### Step 4: Calculate the fundamental frequency using \( f_1 \)
Using \( f_1 \):
\[ f_0 = \frac{105 \, \text{Hz}}{3} = 35 \, \text{Hz} \]

### Step 5: Verify using \( f_2 \) and \( f_3 \)
Using \( f_2 \):
\[ f_0 = \frac{175 \, \text{Hz}}{5} = 35 \, \text{Hz} \]
Using \( f_3 \):
\[ f_0 = \frac{245 \, \text{Hz}}{7} = 35 \, \text{Hz} \]

### Conclusion
The fundamental frequency \( f_0 \) is:
\[ f_0 = 35 \, \text{Hz} \]