While constructing the cumulative frequency column of a frequency distribution, it is noticed that these cumulative frequencies arein arithmetic progression.
Assertion (A) : All the class frequencies are equal.
Reason (R) : when all the class frequencies are equal, the cumulative frequencies are arithmetic progression.

Options

A. Both A and R are individually true, and R is the correct explanation of A.
B. Both A and R are individually true but R is not the correct explanation of A.
C. A is true but R is false.
D. A is false but R is true.

Explanation

To solve the given question, we need to analyze the assertion and the reason provided.

### Step 1: Understand the Assertion
The assertion states that "All the class frequencies are equal." This means that if we have a frequency distribution, each class (or category) has the same frequency.

**Hint**: Think about what it means for frequencies to be equal in a frequency distribution.

### Step 2: Understand the Reason
The reason states that "When all the class frequencies are equal, the cumulative frequencies are in arithmetic progression." Cumulative frequency is calculated by adding the frequencies of all classes up to a certain point. If all class frequencies are equal, the cumulative frequency will increase by the same amount for each class.

**Hint**: Recall how cumulative frequency is calculated and what happens when you add the same number repeatedly.

### Step 3: Calculate Cumulative Frequencies
Let’s assume we have 'n' classes, and each class has a frequency 'f'. The cumulative frequencies (CF) can be calculated as follows:
- For the first class: CF1 = f
- For the second class: CF2 = f + f = 2f
- For the third class: CF3 = f + f + f = 3f
- ...
- For the nth class: CF_n = n * f

This results in cumulative frequencies: f, 2f, 3f, ..., nf.

**Hint**: Write down the cumulative frequencies for a few classes to see the pattern.

### Step 4: Identify the Pattern
The cumulative frequencies form the sequence: f, 2f, 3f, ..., nf. This sequence is an arithmetic progression (AP) where the common difference (d) is 'f'.

**Hint**: Check the definition of an arithmetic progression and see if this sequence fits.

### Step 5: Conclusion
Since we have established that if all class frequencies are equal, the cumulative frequencies indeed form an arithmetic progression, we can conclude that:
- The assertion is true (all class frequencies are equal).
- The reason is also true (equal class frequencies lead to cumulative frequencies in AP).

Thus, both the assertion and the reason are correct, and the reason correctly explains the assertion.

**Final Answer**: Both assertion and reason are individually true, and reason is the correct explanation of assertion.