2014 AMC 10A Problems/Problem 24
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?
If we list the rows by iterations, then we get
so that the th number is the th number on the th row because . 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 .
One may also note we simply need to add the number of skipped numbers to to get our answer. The number of skipped numbers is which has a units digit of . Looking at the answer choices, it becomes apparent that the answer is
Note 2 (A little bit of clarification if the solution wasn't entirely clear at the beginning)
Since it looks kind of weird to pick as the end of the sum, the intuition behind it is that the sum of terms is , since Since we want the th number, we can write our equation as an approximation. Since , , meaning . , so yields , prompting us to lower the value of by around or since we want to lower our answer by and the term below is , which is lower than . Thus, through a little bit of guess and checking, we get This is also similar to the Solution 3, the AOPS video transcript.
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) according to however many numbers are skipped.
Clearly, . This means that there are skipped number "blocks" in the sequence because we started counting from 4.
Therefore , and the answer is .
Solution 3 (AOPS Video Transcript)
First, we group the numbers together in the following way: We quickly realize that the number of terms in the curly braces follow a pattern: (where is the block. Now, we can tell that the last number in a curly brace will be the number of terms in the set added to the number of terms in all the previous sets. Luckily for us, odd numbers are easy to add. If we pretend that there was a at the beginning, then the sum of all of the numbers before and including is . However, we have to subtract which results in . The amount of numbers in the parenthesis are the triangular number or . Next, we want to find the greatest , where . Simplifying, we get . We realize that results in a number just greater than our target. Next, we square : . As we decrease by , we decrease the result of the equation by approximately . In order to decrease by at least , we have to decrease times leading to . We plug it in to getting . This is the last number in the set. The number of terms used is . We need to add terms to get an answer of .
We look at each option starting from . Clearly where is the number of the skipped numbers. So we have . This factors as . Since is a positive integer the answer is .
Video Solution by Richard Rusczyk
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