Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant (common difference, $d$).
  • $n^{\text{th}}$ term: $a_n = a_1 + (n-1)d$
  • Sum of $n$ terms: $S_n = \frac{n}{2}(a_1 + a_n)$ or $S_n = \frac{n}{2}(2a_1 + (n-1)d)$
• Sum of the first $n$ odd natural numbers: $1 + 3 + 5 + \ldots + (2n-1) = n^2$

Step-by-Step Solution

Step 1: Analyze the Structure of the Sets We are given a series of positive multiples of 3 divided into sets: Set 1: $\{3\}$ (1 term) Set 2: $\{6, 9, 12\}$ (3 terms) Set 3: $\{15, 18, 21, 24, 27\}$ (5 terms) We observe that the number of terms in each successive set forms an arithmetic progression: $1, 3, 5, \ldots$. This is an AP with the first term $a_1 = 1$ and a common difference $d = 2$. The number of terms in the $n^{\text{th}}$ set, denoted by $N_n$, can be found using the formula for the $n^{\text{th}}$ term of an AP: $N_n = a_1 + (n-1)d = 1 + (n-1)2 = 1 + 2n - 2 = 2n-1$ For the $11^{\text{th}}$ set ($n=11$), the number of terms is: $N_{11} = 2(11) - 1 = 22 - 1 = 21$ Thus, the $11^{\text{th}}$ set contains 21 terms.

Step 2: Determine the Total Number of Terms Preceding the $11^{\text{th}}$ Set To find the first term of the $11^{\text{th}}$ set, we need to know how many terms are in the first 10 sets. This is the sum of the number of terms in the first 10 sets: $N_1 + N_2 + \ldots + N_{10}$. This sum is the sum of the first 10 odd natural numbers: $1 + 3 + 5 + \ldots + (2(10)-1)$. Using the formula for the sum of the first $n$ odd numbers, $n^2$: Total terms in the first 10 sets = $10^2 = 100$. This means the first 10 sets contain a total of 100 multiples of 3. The $100^{\text{th}}$ multiple of 3 is the last term of the $10^{\text{th}}$ set.

Step 3: Find the First Term of the $11^{\text{th}}$ Set The overall series is the sequence of positive multiples of 3: $3 \times 1, 3 \times 2, 3 \times 3, \ldots$. Since the first 100 terms are in the first 10 sets, the $100^{\text{th}}$ term is $3 \times 100 = 300$. The $11^{\text{th}}$ set begins with the term immediately following the $100^{\text{th}}$ term. Therefore, the first term of the $11^{\text{th}}$ set is the $101^{\text{st}}$ multiple of 3. First term of the $11^{\text{th}}$ set = $3 \times 101 = 303$.

Step 4: Find the Last Term of the $11^{\text{th}}$ Set We know that the $11^{\text{th}}$ set has 21 terms (from Step 1) and its first term is the $101^{\text{st}}$ multiple of 3. The terms in the $11^{\text{th}}$ set are consecutive multiples of 3. The last term will be the $(101 + \text{number of terms} - 1)^{\text{th}}$ multiple of 3. Last term of the $11^{\text{th}}$ set = $(101 + 21 - 1)^{\text{th}}$ multiple of 3 = $121^{\text{st}}$ multiple of 3. Last term of the $11^{\text{th}}$ set = $3 \times 121 = 363$.

Step 5: Calculate the Sum of the Elements in the $11^{\text{th}}$ Set The $11^{\text{th}}$ set forms an arithmetic progression with:

• Number of terms ($n$) = 21
• First term ($a_1$) = 303
• Last term ($a_n$) = 363 Using the sum formula for an AP, $S_n = \frac{n}{2}(a_1 + a_n)$: Sum of elements in the $11^{\text{th}}$ set = $\frac{21}{2}(303 + 363)$ Sum = $\frac{21}{2}(666)$ Sum = $21 \times 333$ Sum = $6993$.

Common Mistakes & Tips

• Confusing Indices: Be careful to distinguish between the index of a set (e.g., the $11^{\text{th}}$ set) and the index of a term within the overall sequence of multiples of 3.
• Number of Terms Pattern: Correctly identifying the pattern of the number of terms in each set ($1, 3, 5, \ldots$) is crucial for determining the size of the target set and the cumulative count of preceding terms.
• Sum of Odd Numbers Formula: Remembering that the sum of the first $n$ odd numbers is $n^2$ provides a quick way to calculate the total number of terms before a given set.

Summary To find the sum of elements in the $11^{\text{th}}$ set, we first determined the number of terms in the $11^{\text{th}}$ set by recognizing the pattern of odd numbers for the sizes of successive sets. Then, we calculated the total number of terms in the preceding 10 sets. This allowed us to find the first term of the $11^{\text{th}}$ set as the subsequent multiple of 3. Finally, knowing the first term, the number of terms, and that the terms are consecutive multiples of 3, we identified the last term and applied the arithmetic progression sum formula to find the total sum.

The final answer is $\boxed{6993}$.