Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant (common difference, $d$). The $n$-th term is $a_n = a_1 + (n-1)d$.
• Sum of an AP: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$.
• Common Terms of Two APs: The terms common to two APs themselves form an AP. The common difference of this new AP is the Least Common Multiple (LCM) of the common differences of the original APs.

Step-by-Step Solution

Step 1: Analyze the Given Arithmetic Progressions We are given two arithmetic progressions. Let's identify their first term, common difference, and last term.

• First AP ($AP_1$): $3, 7, 11, 15, \ldots, 403$

  • First term ($a_1$) = 3
  • Common difference ($d_1$) = $7 - 3 = 4$
  • Last term ($L_1$) = 403

• Second AP ($AP_2$): $2, 5, 8, 11, \ldots, 404$

  • First term ($a_2$) = 2
  • Common difference ($d_2$) = $5 - 2 = 3$
  • Last term ($L_2$) = 404

Step 2: Find the First Common Term A term that is common to both APs must be of the form $a_1 + (k-1)d_1$ and $a_2 + (l-1)d_2$ for some positive integers $k$ and $l$. Let the common term be $T$. Then, $T = 3 + (k-1)4$ $T = 2 + (l-1)3$

Equating these, we get: $3 + 4k - 4 = 2 + 3l - 3$ $4k - 1 = 3l - 1$ $4k = 3l$

Since 4 and 3 are coprime, for $4k$ to be equal to $3l$, $k$ must be a multiple of 3, and $l$ must be a multiple of 4. To find the first common term, we need the smallest positive integer values for $k$ and $l$. The smallest positive integer for $k$ that is a multiple of 3 is $k=3$. Substituting $k=3$ into $4k = 3l$: $4(3) = 3l \implies 12 = 3l \implies l=4$.

Now, we find the value of this common term using either AP: Using $AP_1$ with $k=3$: $T = 3 + (3-1)4 = 3 + 2 \times 4 = 3 + 8 = 11$. Using $AP_2$ with $l=4$: $T = 2 + (4-1)3 = 2 + 3 \times 3 = 2 + 9 = 11$. So, the first common term is $A = 11$.

Step 3: Determine the Common Difference of the AP of Common Terms The common terms of two APs form an AP themselves. The common difference of this new AP ($D$) is the LCM of the common differences of the original APs. $d_1 = 4$ and $d_2 = 3$. $D = \text{LCM}(d_1, d_2) = \text{LCM}(4, 3)$. Since 4 and 3 are coprime, their LCM is their product: $D = 4 \times 3 = 12$. The AP of common terms starts with 11 and has a common difference of 12.

Step 4: Find the Last Common Term and the Number of Common Terms The common terms must be present in both original APs. Therefore, a common term cannot exceed the minimum of the last terms of the two APs. Minimum of last terms = $\min(L_1, L_2) = \min(403, 404) = 403$.

Let the AP of common terms be $A_n = A + (n-1)D$. We need to find the largest integer $n$ such that $A_n \le 403$. $A_n = 11 + (n-1)12$. So, we set: $11 + (n-1)12 \le 403$ Subtract 11 from both sides: $(n-1)12 \le 403 - 11$ $(n-1)12 \le 392$ Divide by 12: $n-1 \le \frac{392}{12}$ $n-1 \le \frac{98}{3}$ $n-1 \le 32.66\ldots$

Since $n-1$ must be an integer, the largest possible value for $n-1$ is 32. $n-1 = 32 \implies n = 33$. Thus, there are $N=33$ common terms.

The last common term ($A_{33}$) is: $A_{33} = 11 + (33-1)12 = 11 + 32 \times 12 = 11 + 384 = 395$. This term 395 is indeed $\le 403$.

Step 5: Calculate the Sum of the Common Terms We have the first common term ($A = 11$), the last common term ($A_{33} = 395$), and the number of common terms ($N=33$). We use the sum formula for an AP: $S_N = \frac{N}{2}(A + A_N)$ $S_{33} = \frac{33}{2}(11 + 395)$ $S_{33} = \frac{33}{2}(406)$ $S_{33} = 33 \times 203$ $S_{33} = 6699$.

Common Mistakes & Tips

• Incorrect Last Common Term: Ensure the last common term does not exceed the minimum of the last terms of the original APs.
• LCM Calculation: Accurately calculate the LCM of the common differences to find the common difference of the common terms.
• Integer Indices: When solving for the number of terms ($n$), remember that $n$ and $n-1$ must be integers. Round down if necessary when finding the maximum number of terms.

Summary The problem involved finding the sum of terms common to two given arithmetic progressions. We first identified the properties of each AP. Then, we determined the first common term by setting the general terms equal and finding the smallest integer indices. The common difference of the AP formed by the common terms was found by taking the LCM of the original common differences. Finally, we identified the last common term by considering the minimum of the last terms of the original APs and calculated the total number of common terms. Using these values, the sum of the common terms was computed using the AP sum formula. The first common term is 11, the common difference of the common terms is 12, and there are 33 common terms, with the last being 395. The sum of these terms is 6699.

The final answer is \boxed{6699}.