Key Concepts and Formulas

• Arithmetic Progression (AP): A sequence where the difference between consecutive terms is constant.
  • First term: $a$
  • Common difference: $d$
  • $n$-th term: $a_n = a + (n-1)d$
  • Sum of the first $n$ terms: $S_n = \frac{n}{2}[2a + (n-1)d]$
• Constraints: The problem specifies an AP of "positive integers". This means $a \in \mathbb{Z}^+$, $d \in \mathbb{Z}$, and $a_n > 0$ for all relevant $n$.

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Step-by-Step Solution

Step 1: Use the sum of the first three terms to establish a relationship between $a$ and $d$. We are given that the sum of the first three terms ($S_3$) is 54. Using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ with $n=3$: $S_3 = \frac{3}{2}[2a + (3-1)d]$ $54 = \frac{3}{2}[2a + 2d]$ $54 = \frac{3}{2} \cdot 2(a+d)$ $54 = 3(a+d)$ Dividing by 3, we get: $18 = a+d$ This implies $a = 18-d$.

Step 2: Apply the "positive integers" constraint to find bounds for $d$. Since the AP consists of positive integers, the first term $a$ must be a positive integer ($a > 0$). Substituting $a = 18-d$: $18-d > 0 \implies d < 18$ Also, all terms must be positive. Specifically, the 20th term ($a_{20}$) must be positive. $a_{20} = a + (20-1)d = a + 19d$ Substitute $a = 18-d$: $a_{20} = (18-d) + 19d = 18 + 18d$ For $a_{20} > 0$: $18 + 18d > 0$ $18d > -18$ $d > -1$ Since $a$ and $d$ are integers, we have $d \in \mathbb{Z}$ and $-1 < d < 18$. This means $d$ can be any integer from 0 to 17, inclusive.

Step 3: Use the sum of the first twenty terms to form an inequality for $d$. We are given that the sum of the first twenty terms ($S_{20}$) lies between 1600 and 1800: $1600 < S_{20} < 1800$ Using the formula $S_n = \frac{n}{2}[2a + (n-1)d]$ with $n=20$: $S_{20} = \frac{20}{2}[2a + (20-1)d]$ $S_{20} = 10[2a + 19d]$ Substitute $a = 18-d$ into the expression for $S_{20}$: $S_{20} = 10[2(18-d) + 19d]$ $S_{20} = 10[36 - 2d + 19d]$ $S_{20} = 10[36 + 17d]$ Now, substitute this into the given inequality: $1600 < 10[36 + 17d] < 1800$ Divide all parts by 10: $160 < 36 + 17d < 180$ Subtract 36 from all parts: $160 - 36 < 17d < 180 - 36$ $124 < 17d < 144$ Divide all parts by 17: $\frac{124}{17} < d < \frac{144}{17}$ Calculating the decimal values: $7.294... < d < 8.470...$

Step 4: Determine the unique value of $d$ by combining the constraints. We have two conditions for $d$:

1. $d$ is an integer such that $-1 < d < 18$.
2. $d$ satisfies $7.294... < d < 8.470...$.

The only integer that lies strictly between 7.294... and 8.470... is $d=8$. This value also satisfies the condition $-1 < d < 18$. Thus, the common difference is $d=8$.

Step 5: Calculate the first term $a$. Using the relationship $a = 18-d$ from Step 1: $a = 18 - 8$ $a = 10$ Since $a=10$ is a positive integer, this is a valid first term.

Step 6: Calculate the 11th term of the AP. We need to find $a_{11}$ using the formula $a_n = a + (n-1)d$ with $n=11$, $a=10$, and $d=8$: $a_{11} = 10 + (11-1) \cdot 8$ $a_{11} = 10 + 10 \cdot 8$ $a_{11} = 10 + 80$ $a_{11} = 90$

Common Mistakes & Tips

• Integer Constraints: Remember that $a$ and $d$ must be integers, and all terms must be positive. This can significantly narrow down possibilities, especially for $d$.
• Inequality Handling: When solving inequalities, ensure all operations are applied to all parts of the inequality to maintain its validity.
• Formula Accuracy: Double-check the formulas for the $n$-th term and the sum of an AP, particularly the $(n-1)$ factor for the common difference.

Summary

We utilized the given information about the sum of the first three terms to establish a linear relationship between the first term ($a$) and the common difference ($d$). The constraint that the AP consists of positive integers provided crucial bounds for $d$. Subsequently, the information about the sum of the first twenty terms was used to form an inequality, which, when combined with the integer constraints on $d$, uniquely determined $d=8$. With $d$ found, we calculated $a=10$. Finally, we computed the 11th term using the standard AP formula.

The final answer is $\boxed{90}$.