Animated Solution for Mathematics - Sequence and Series: Consider an A. P. of positive integers, whose sum of the first three terms is 54 and the sum of the first twenty terms lies between 1600 and 1800. Then its $11^{th}$ term is :

Imagine you are standing at the start of a sequence of numbers. You know that each step you take, you add the same amount to get to the next number. This is the elegance of an Arithmetic Progression (A.P.).

In this problem, we are dealing with a sequence of positive integers. This is a crucial piece of information; it is not just flavor text, but a mathematical constraint that acts as our compass, guiding us through the algebraic landscape.

We begin by defining our A.P. with a first term $a$ and a common difference $d$. The problem states that all terms are positive integers.

This implies that both $a$ and $d$ must be integers. If $a$ is an integer and $a+d$ is an integer, then $d$ must also be an integer. Keep this in mind; it is the key that will unlock the final answer.

We are given that the sum of the first three terms is $54$. Using the sum formula $S_n=\frac{n}{2}[2a+(n-1)d]$, we can write:

Simplifying this, we get $3(a+d)=54$, which leads us to the relation:

This means we can express our first term as $a=18-d$. We have successfully reduced our problem from two variables to one.

Now, let us look at the sum of the first twenty terms, $S_{20}$. Using the formula again:

By substituting our expression $a=18-d$ into this, we get:

The problem states that $1600<S_{20}<1800$. Substituting our expression for $S_{20}$, we get the inequality:

Dividing by $10$, we have $160<36+17d<180$. Subtracting $36$ from all parts, we get:

Dividing by $17$, we find that $7.29<d<8.47$. Since we established that $d$ must be an integer, the only possible value for $d$ is $8$.

With $d=8$, we can easily find $a=18-8=10$. Finally, we calculate the $11^{th}$ term using the formula $T_n=a+(n-1)d$:

The journey from the initial constraints to the final result is a testament to how algebraic manipulation, when guided by logical constraints, leads us directly to the truth. The final answer is 90.