2014 AMC 10A Problem 24

Question

A sequence of natural numbers is constructed by listing the first

4,

then skipping one, listing the next

5,

skipping

2,

listing

6,

skipping

3,

and on the

n

th iteration, listing

n+3

and skipping

n.

The sequence begins

1,2,3,4,6,7,8,9,10,13.

What is the

500{,}000

th number in the sequence?

996,

\!506

996,

\!507

996,

\!508

996,

\!509

996,

\!510

Count how many terms have been listed after

n

full iterations

Then locate the desired term inside the next listed block

After

n

full iterations, the number of listed terms is

4+5+\cdots+(n+3)=\frac{n(n+7)}2

.

We need the largest

n

with

\frac{n(n+7)}2<500000

. Since

996\cdot1003=998988

, after

996

iterations there are

499494

listed numbers.

The first number listed in iteration

997

is one more than the total of all listed and skipped numbers so far, namely

996^2+4\cdot996+1=996001

.

The

500000

th listed number is the

500000-499494=506

th number of this next block, so it is

996001+505=996506

.

Thus, A is the correct answer.

Problem 24 in other years