Key Concepts and Formulas

This problem requires understanding and applying concepts related to Arithmetic Progressions (APs). The key formulas needed are:

1. General Term of an AP: The $n$-th term of an AP with first term $a$ and common difference $d$ is given by $t_n = a + (n-1)d$.
2. Sum of n terms of an AP: The sum of the first $n$ terms is $S_n = \frac{n}{2}[2a + (n-1)d]$.
3. Common Terms of Two APs: If two APs have common differences $d_1$ and $d_2$, the series formed by their common terms is also an AP. The common difference of this new AP is the Least Common Multiple (LCM) of $d_1$ and $d_2$, i.e., $d_{\text{common}} = \text{LCM}(d_1, d_2)$.

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Step-by-Step Solution

Step 1: Analyze the Given Arithmetic Progressions

We are given two series, which are arithmetic progressions. Let's identify their first term, common difference, and the total number of terms. We will also calculate their last terms, as common terms cannot extend beyond the last term of either series.

• First Series: $3, 6, 9, 12, \ldots$ upto 78 terms

  • First term ($a_1$) = 3
  • Common difference ($d_1$) = $6 - 3 = 3$
  • Number of terms ($N_1$) = 78
  • Last term ($t_{78}$): Using $t_n = a + (n-1)d$ $t_{78} = 3 + (78 - 1)3 = 3 + 77 \times 3 = 3 + 231 = 234$ The first series is $3, 6, 9, \ldots, 234$.

• Second Series: $5, 9, 13, 17, \ldots$ upto 59 terms

  • First term ($a_2$) = 5
  • Common difference ($d_2$) = $9 - 5 = 4$
  • Number of terms ($N_2$) = 59
  • Last term ($t_{59}$): Using $t_n = a + (n-1)d$ $t_{59} = 5 + (59 - 1)4 = 5 + 58 \times 4 = 5 + 232 = 237$ The second series is $5, 9, 13, \ldots, 237$.

Step 2: Determine the First Common Term

To find the first term of the AP formed by common terms, we need to find the smallest number that appears in both series.

• First Series terms: $3, 6, \mathbf{9}, 12, 15, 18, 21, \ldots$
• Second Series terms: $5, \mathbf{9}, 13, 17, 21, 25, \ldots$ By inspection, the first common term is 9. Let's denote the first term of the common AP as $A$. So, $A = 9$.

Step 3: Determine the Common Difference of the Common Terms AP

The common difference of the AP formed by common terms is the LCM of the common differences of the original APs.

• Common difference of the first AP ($d_1$) = 3
• Common difference of the second AP ($d_2$) = 4

The common difference of the new AP, let's call it $D$, is: $D = \text{LCM}(d_1, d_2) = \text{LCM}(3, 4)$ Since 3 and 4 are coprime, their LCM is their product: $D = 3 \times 4 = 12$

Step 4: Formulate the AP of Common Terms

With the first term $A=9$ and common difference $D=12$, the AP of common terms is: $9, (9+12), (9+2\times12), \ldots$ $9, 21, 33, \ldots$

Step 5: Determine the Number of Common Terms

The common terms must be present in both original series. This means any common term must be less than or equal to the last term of both original series.

• Last term of the first series = 234
• Last term of the second series = 237

So, any common term must be less than or equal to $\min(234, 237) = 234$. Let $n$ be the number of common terms. The $n$-th term of the common AP is given by $t_n = A + (n-1)D$. We must have: $9 + (n-1)12 \le 234$ Subtract 9 from both sides: $(n-1)12 \le 234 - 9$ $(n-1)12 \le 225$ Divide by 12: $n-1 \le \frac{225}{12}$ $n-1 \le 18.75$ Add 1 to both sides: $n \le 18.75 + 1$ $n \le 19.75$ Since $n$ must be an integer, the maximum possible value for $n$ is 19. Therefore, there are 19 common terms.

Step 6: Calculate the Sum of the Common Terms

We need to find the sum of these 19 common terms. Using the sum formula $S_n = \frac{n}{2}[2A + (n-1)D]$ with $n=19$, $A=9$, and $D=12$: $S_{19} = \frac{19}{2}[2 \times 9 + (19-1)12]$ $S_{19} = \frac{19}{2}[18 + (18)12]$ $S_{19} = \frac{19}{2}[18 + 216]$ $S_{19} = \frac{19}{2}[234]$ Divide 234 by 2: $S_{19} = 19 \times 117$ Calculating the product: $19 \times 117 = 2223$ Thus, the sum of the terms common to both series is 2223.

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Common Mistakes & Tips

• Identifying the first common term: Ensure you don't just pick the smallest term from either series; verify it is present in the other series.
• Common difference of common terms: Always use the LCM of the individual common differences, not the GCD or sum/difference.
• Upper bound for common terms: The number of common terms is limited by the smaller of the last terms of the original two series.

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Summary

We analyzed the two given arithmetic progressions, determined their first terms, common differences, and last terms. We found the first common term by inspection and calculated the common difference of the series of common terms using the LCM of the individual common differences. We then determined the maximum number of common terms by considering the last terms of the original series. Finally, we applied the sum formula for an arithmetic progression to find the sum of these common terms. The sum of the terms common to both series is 2223.

The final answer is $\boxed{2223}$.