The first term in a sequence is \(24\). We can determine the next terms in the sequence as follows:
If a term is even, then divide it by \(2\) to get the next term.
If a term is odd, then multiply it by \(3\) and add \(1\) to get the next term.
By doing this, we can determine that the first three terms in the sequence are \(24,\) \(12,\) and \(6.\)
Elias writes the first \(n\) terms in this sequence and notices that the sum of these terms is a four-digit number. What is the smallest possible value of \(n\)?
We will begin by finding more terms in the sequence. The first \(14\) terms of the sequence are \(24, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1.\)
If we continue the sequence, we will see that the terms \(4,\) \(2,\) and \(1\) will continue to repeat. Now we want to find the smallest possible value of \(n\) so that the sum of the terms in the sequence from term \(1\) to term \(n\) is at least \(1000\).
The sum of the first \(8\) terms is \(24+12+6+3+10+5+16+8=84\). The sum of the repeating numbers is \(4+2+1=7\). We want to determine the number of groups of repeating numbers. Let this be \(g\). Suppose \(84+7g = 1000\). Solving this gives \(7g = 916,\) so \(g \approx 130.857.\)
If \(g=130,\) then the sum of the terms in the sequence is \(84 + 7 \times 130 = 994\). This sequence contains the first \(8\) terms, plus \(130\) groups of the three repeating numbers. Therefore there are a total of \(8+3 \times 130 = 398\) terms.
The 399th term in the sequence will be \(4,\) so the sum of the first \(399\) terms will be \(994+4=998\).
The 400th term in the sequence will be \(2,\) so the sum of the first \(400\) terms will be \(998+2=1000\). This is the smallest possible four-digit number, so the smallest possible value of \(n\) is \(400\).
Extension:
In \(1937,\) the mathematician Lothat Collatz wondered if any sequence whose terms after the first are determined in this way would always eventually reach the number \(1,\) regardless of which number you started with. This problem is actually still unsolved today and is called the Collatz Conjecture.