The average of the first seven terms of an AP is \({3 \over {7}}\) times the average of the 7th to 13th terms of the same AP. The average of the first four terms will be

Given that the average of the first seven terms of an arithmetic progression (AP) is \(\frac{3}{7}\) times the average of the 7th to 13th terms, we are required to find the average of the first four terms.

### Step 1: Express the terms of the AP
Let the first term of the AP be \(a\) and the common difference be \(d\).

- The first seven terms are \(a, a+d, a+2d, \dots, a+6d\).
- The 7th to 13th terms are \(a+6d, a+7d, a+8d, \dots, a+12d\).

### Step 2: Calculate the average of the first seven terms
The average of the first seven terms is:
\[
\text{Average}_1 = \frac{a + (a+d) + (a+2d) + \cdots + (a+6d)}{7} = \frac{7a + 21d}{7} = a + 3d
\]

### Step 3: Calculate the average of the 7th to 13th terms
The average of the 7th to 13th terms is:
\[
\text{Average}_2 = \frac{(a+6d) + (a+7d) + \cdots + (a+12d)}{7} = \frac{7a + 63d}{7} = a + 9d
\]

### Step 4: Use the given condition
According to the problem, the average of the first seven terms is \(\frac{3}{7}\) times the average of the 7th to 13th terms:
\[
a + 3d = \frac{3}{7} \times (a + 9d)
\]
Multiply both sides by 7 to eliminate the fraction:
\[
7a + 21d = 3a + 27d
\]
Simplify by subtracting \(3a\) and \(21d\) from both sides:
\[
4a = 6d \quad \Rightarrow \quad a = \frac{3d}{2}
\]

### Step 5: Calculate the average of the first four terms
The first four terms are \(a, a+d, a+2d, a+3d\). The average of these terms is:
\[
\text{Average}_3 = \frac{a + (a+d) + (a+2d) + (a+3d)}{4} = \frac{4a + 6d}{4} = a + \frac{3d}{2}
\]
Substituting \(a = \frac{3d}{2}\) into the equation:
\[
\text{Average}_3 = \frac{3d}{2} + \frac{3d}{2} = 3d
\]

Thus, the average of the first four terms is \( \boxed{3d} \).

Would you like more details or have any questions?

Here are some related questions to expand your understanding:
1. How would the solution change if the ratio was different, say \(\frac{5}{7}\)?
2. What is the sum of the first \(n\) terms of the AP given in the problem?
3. How does the common difference \(d\) influence the averages of different segments of the AP?
4. What if the average of the first \(n\) terms was given directly instead of the ratio?
5. Can this method be extended to non-integer sequences, and how would that affect the result?

**Tip:** When dealing with arithmetic progressions, expressing terms and averages in terms of the first term and common difference often simplifies solving the problem.

The average of the first seven terms of an AP is \({3 \over 7}\) times the average of the 7th to 13th terms of the same AP. The average of the first four terms will be

Find the average of the first four terms of an arithmetic progression (AP) when the average of the first seven terms is 3/7 times the average of the 7th to 13th terms. This problem involves understanding the properties of APs and applying formulas to solve linear equations. Suitable for Grades 10-12.

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