Difference between revisions of "2014 AMC 10A Problems/Problem 24"
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==Problem== | ==Problem== | ||
− | A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,000</math>th number in the sequence? | + | A sequence of natural numbers is constructed by listing the first <math>4</math>, then skipping one, listing the next <math>5</math>, skipping <math>2</math>, listing <math>6</math>, skipping <math>3</math>, and, on the <math>n</math>th iteration, listing <math>n+3</math> and skipping <math>n</math>. The sequence begins <math>1,2,3,4,6,7,8,9,10,13</math>. What is the <math>500,\!000</math>th number in the sequence? |
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+ | <math> \textbf{(A)}\ 996,\!506\qquad\textbf{(B)}\ 996,\!507\qquad\textbf{(C)}\ 996,\!508\qquad\textbf{(D)}\ 996,\!509\qquad\textbf{(E)}\ 996,\!510 </math> | ||
==Solution 1== | ==Solution 1== |
Revision as of 19:03, 1 February 2015
Contents
[hide]Problem
A sequence of natural numbers is constructed by listing the first , then skipping one, listing the next
, skipping
, listing
, skipping
, and, on the
th iteration, listing
and skipping
. The sequence begins
. What is the
th number in the sequence?
Solution 1
If we list the rows by iterations, then we get
etc.
so that the th number is the
th number on the
th row. (
) The last number of the
th row (when including the numbers skipped) is
, (we add the
because of the numbers we skip) so our answer is
Solution 2
Let's start with natural numbers, with no skips in between.
All we need to do is count how many numbers are skipped, , and "push" (add on to)
however many numbers are skipped.
Clearly, . This means that the number skipped number "blocks" in the sequence is
because we started counting from 4.
Therefore , and the answer is
.
See Also
2014 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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