Consider a decreasing sequence of $n$ positive integers $x_1>x_2>\cdots>x_n$ that satisfies the following conditions:
The average of the first $3$ terms in the sequence is $2025.$
For all $4 \lt k\lt n,$ the average of the first $k$ terms is $1$ less than the average of the first $k - 1$ terms.
What is the greatest possible value of ${n?}$
$\textbf{(A) } 1013 \qquad \textbf{(B) } 1014 \qquad \textbf{(C) } 1016 \qquad \textbf{(D) } 2016 \qquad \textbf{(E) } 2025$
Difficulty: 32/100 - Another application of arithmetic means hidden in a sequence.
Core Concepts: arithmetic mean, sequence, optimization, recursion
Challenges: Converting the mean condition into a more manageable equation.
Techniques: Turning means into sums, execute a recursion to find a maximum value for $x_n$.
Error-prone Steps: Mishandling the index position for the sequence.
Ideal Time:
Experienced: ≤ 1-2 min
Intermediate: ≤ 3 min
Beginner: ≤ 6-7 min