Question

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \le n \le 200.$ $1,$ $2, 2,$ $3, 3, 3,$ $4, 4, 4, 4,$ $\ldots,$ $200, 200,$ $\ldots, 200$

What is the median of the numbers in this list?

$100.5$

$134$

$142$

$150.5$

$167$

Answer: C

Concepts:triangular number median (data)

Difficulty rating: 1730

Solution:

The list has $1 + 2 + \cdots + 200$ $= \dfrac{200\cdot 201}{2}$ $= 20100$ terms, so the median is the average of the $10050$ th and $10051$ st terms.

The value $n$ occupies positions up to $\dfrac{n(n+1)}{2}.$ Since $\dfrac{141\cdot 142}{2} = 10011$ and $\dfrac{142\cdot 143}{2} = 10153,$ positions $10012$ through $10153$ all equal $142.$ Both middle positions fall in this block, so the median is $142.$

Thus, the correct answer is C.

2021 AMC 12A Spring Problem 16

Answer: C

Difficulty rating: 1730

Solution:

Problem 16 in Other Years