Question

The sum of $25$ consecutive even integers is $10{,}000.$ What is the largest of these $25$ consecutive even integers?

$360$

$388$

$412$

$416$

$424$

Answer: E

Concepts:arithmetic sequence

Difficulty rating: 1100

Small Hint:

For an odd number of consecutive terms, the average is the middle term.

Big Hint:

There are $12$ even integers larger than the middle term.

Video solution:

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Written solution:

The average of these numbers is $\frac{10{,}000}{25} = 400.$ The largest number is $12$ even numbers away, which means that it equals $400 + 12 \cdot 2 = 424.$

Thus, E is the correct answer.

2016 AMC 8 Problem 19