Let `3, 7, 11, 15, ...., 403` and `2, 5, 8, 11, . . ., 404` be two arithmetic progressions. Then the sum, of the common terms in them, is equal to _________.

Answer

To find the sum of the common terms in the two arithmetic progressions (APs), we can follow these steps:

### Step 1: Identify the two APs
The first AP is:
- \( a_1 = 3, a_2 = 7, a_3 = 11, \ldots, a_n = 403 \)
This AP has:
- First term \( a = 3 \)
- Common difference \( d_1 = 4 \)

The second AP is:
- \( b_1 = 2, b_2 = 5, b_3 = 8, \ldots, b_m = 404 \)
This AP has:
- First term \( b = 2 \)
- Common difference \( d_2 = 3 \)

### Step 2: Find the common terms
To find the common terms, we need to find the terms that appear in both sequences. The common terms will also form an AP.

The first common term can be found by checking the terms of both APs. The first common term is \( 11 \).

### Step 3: Determine the common difference of the common terms
The common difference of the common terms can be found using the least common multiple (LCM) of the two common differences:
- \( d_1 = 4 \)
- \( d_2 = 3 \)

The LCM of \( 4 \) and \( 3 \) is \( 12 \). Thus, the common difference \( d_c = 12 \).

### Step 4: Write the general term of the common AP
The common terms can be expressed as:
- \( T_n = 11 + (n - 1) \cdot 12 \)

### Step 5: Find the maximum value of \( n \) such that \( T_n \leq 403 \)
Set up the inequality:
\[ 11 + (n - 1) \cdot 12 \leq 403 \]
Solving for \( n \):
\[ (n - 1) \cdot 12 \leq 392 \]
\[ n - 1 \leq \frac{392}{12} \approx 32.67 \]
Thus, \( n \leq 33.67 \), meaning \( n \) can take integer values up to \( 33 \).

### Step 6: Calculate the sum of the first \( n \) common terms
The number of common terms is \( n = 33 \). We can use the formula for the sum of an AP:
\[ S_n = \frac{n}{2} \cdot (2a + (n - 1)d) \]
Where:
- \( a = 11 \)
- \( d = 12 \)

Substituting the values:
\[ S_{33} = \frac{33}{2} \cdot (2 \cdot 11 + (33 - 1) \cdot 12) \]
\[ = \frac{33}{2} \cdot (22 + 32 \cdot 12) \]
\[ = \frac{33}{2} \cdot (22 + 384) \]
\[ = \frac{33}{2} \cdot 406 \]
\[ = 33 \cdot 203 \]
\[ = 6699 \]

### Step 7: Final answer
The sum of the common terms in the two arithmetic progressions is \( 6699 \).

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