Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant. The sum of the first $n$ terms of an AP is given by $S_n = \frac{n}{2}[2a + (n-1)d]$, where $a$ is the first term and $d$ is the common difference.
• Pattern Recognition in Series: Identifying the underlying pattern of a series is crucial for determining the appropriate method for calculating its sum, especially when it's not a standard AP or GP. Grouping terms can often simplify complex series.

Step-by-Step Solution

Step 1: Analyze the given series and identify its pattern. The given series is $3 + 4 + 8 + 9 + 13 + 14 + 18 + 19 + \ldots$ for 40 terms. Let's examine the differences between consecutive terms: $4 - 3 = 1$ $8 - 4 = 4$ $9 - 8 = 1$ $13 - 9 = 4$ $14 - 13 = 1$ $18 - 14 = 4$ $19 - 18 = 1$ The differences alternate between $1$ and $4$. This indicates that the series is not a simple Arithmetic Progression. The repeating pattern of differences suggests that we can group terms to simplify the series.

Why this step? Understanding the nature of the series is the first and most critical step. Directly applying AP or GP formulas without verifying the pattern can lead to incorrect solutions. Observing the alternating differences is key to unlocking the problem's structure.

Step 2: Form a new series by grouping terms. Since the pattern of differences is $(+1, +4)$, we can group the terms of the original series in pairs: $(3 + 4) + (8 + 9) + (13 + 14) + (18 + 19) + \ldots$ Calculating the sum of each pair gives us a new series: $7 + 17 + 27 + 37 + \ldots$

Why this step? The original series is not an AP. By grouping terms into pairs, we aim to transform it into a simpler, recognizable sequence, which is likely an AP, allowing us to use standard summation formulas.

Step 3: Determine the number of terms in the new series. The original series has 40 terms. Since we grouped these terms into pairs, and each pair forms one term in the new series, the number of terms in the new series will be half the number of terms in the original series. Number of terms in the new series, $n = \frac{\text{Total terms in original series}}{2}$ $n = \frac{40}{2} = 20$ terms.

Why this step? It is essential to correctly identify the number of terms ($n$) for the new series we have formed. Using the original number of terms (40) would lead to an incorrect sum calculation.

Step 4: Identify the parameters of the new Arithmetic Progression. The new series is $7 + 17 + 27 + 37 + \ldots$ with 20 terms. Let's confirm if this new series is an AP: $17 - 7 = 10$ $27 - 17 = 10$ $37 - 27 = 10$ The common difference is constant ($10$), so this is an Arithmetic Progression. The parameters of this AP are:

• First term ($a$) = $7$.
• Common difference ($d$) = $10$.
• Number of terms ($n$) = $20$.

Why this step? These three parameters ($a$, $d$, and $n$) are necessary inputs for applying the Arithmetic Progression sum formula. Correctly identifying them ensures the subsequent calculation is accurate.

Step 5: Apply the sum formula for an Arithmetic Progression. We use the formula for the sum of the first $n$ terms of an AP: $S_n = \frac{n}{2}[2a + (n-1)d]$. Substituting the values $a=7$, $d=10$, and $n=20$: $S_{20} = \frac{20}{2}[2(7) + (20-1)10]$ $S_{20} = 10[14 + (19)10]$ $S_{20} = 10[14 + 190]$ $S_{20} = 10[204]$ $S_{20} = 2040$ This is the sum of the first 40 terms of the original series.

Why this step? This calculation directly yields the sum of the first 40 terms of the given series, which is required to solve the problem.

Step 6: Compare the calculated sum with the given expression and find 'm'. The problem states that the sum of the first 40 terms of the series is $(102)m$. We have calculated this sum to be $2040$. Equating the two expressions: $2040 = (102)m$ To find $m$, divide both sides by $102$: $m = \frac{2040}{102}$ $m = 20$

Why this step? This is the final step where we use our calculated sum to find the value of $m$ as requested by the question, by setting up and solving an equation.

Common Mistakes & Tips

• Misidentifying the Series Type: Do not assume a series is an AP or GP without checking the differences or ratios. The alternating differences in this problem are a clear indicator that it's not a simple AP.
• Incorrectly Adjusting 'n': When grouping terms, always recalculate the number of terms ($n$) for the new, simplified series. Using the original number of terms is a common error.
• Calculation Errors: Be meticulous with arithmetic, especially when substituting values into the sum formula.

Summary

The given series has a pattern of alternating differences, suggesting that grouping terms in pairs would simplify it into an Arithmetic Progression. By pairing consecutive terms, we formed a new AP with the first term $a=7$, common difference $d=10$, and $n=20$ terms. The sum of this AP was calculated to be $2040$. We then equated this sum to the given expression $(102)m$ and solved for $m$, finding $m=20$.

The final answer is $\boxed{20}$ which corresponds to option (A).