Sequences & Series
Sequence and Series
Allen Star Batch
Grade 11
Question:
Integers $1, 2, 3, ..., n$ where $n > 2$ are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed and the average of the remaining numbers is found to be 17. What is the maximum sum of the two removed numbers ?
Step-by-Step Solution
Key Concept: Rearrange the constraint into a quadratic inequality in $n$ to find the feasible range, then optimize $m+k$ within that range.
From $\frac{n(n+1)}{2} - (m+k) = 17(n-2)$, we derive $n^2 - 33n + 68 = 2(m+k)$. This gives $n \in [31, 35]$ with $n = \{31, 32, 33, 34, 35\}$. The maximum value of $m + k$ occurs when $n = 35$, yielding $m + k = 35 + 34 = 69$.
Correct Answer: 69