Question

consider the situation, on day 1, jaime bought a copy of a song and played it for 6 of his friends. on day 2, 4 of those 6 people bought a copy of the song, and each of them played the song for 6 of their friends. on day 3, 4 people out of each group of 6 people from day 2 bought a copy of the song and played it for 6 of their friends. assume this pattern continues. drag the needed numbers and operations to write a recursive function for the number of people who buy the song on day ( a_1 = ) ( a_n = ) 0 ( \frac{4}{6} ) 1 2 4 6 8 ( a_{n - 1} ) ( a_{n + 1} ) + ×

Explanation:

Step1: Determine \(a_1\)

On day 1, Jaime bought the song, so the number of people who bought the song on day 1 is 1. Thus, \(a_1 = 1\).

Step2: Determine the recursive formula \(a_n\)

On day 2, 4 out of 6 people from day 1's group (which is \(a_1\)) bought the song. Wait, actually, on day \(n\), the number of people who bought the song should be related to the number of people who bought it on day \(n - 1\). Each person who bought the song on day \(n-1\) played it for 6 friends, and 4 out of those 6 bought it. So the number of people who buy on day \(n\) is \(a_n=a_{n - 1}\times\frac{4}{6}\times6\)? Wait, no. Wait, on day 1, \(a_1 = 1\). On day 2, the 1 person (Jaime) played it for 6 friends, 4 of them bought it, so \(a_2=4\). On day 3, each of the \(a_2 = 4\) people played it for 6 friends, and 4 out of 6 bought it, so \(a_3=4\times\frac{4}{6}\times6\)? No, that's not right. Wait, no, the number of people who buy on day \(n\): on day \(n-1\), \(a_{n - 1}\) people bought it. Each of those \(a_{n - 1}\) people played it for 6 friends, so there are \(a_{n - 1}\times6\) friends. Then 4 out of 6 of those friends buy it, so \(a_n=a_{n - 1}\times6\times\frac{4}{6}=a_{n - 1}\times4\)? Wait, but on day 1, \(a_1 = 1\), day 2: \(1\times6\times\frac{4}{6}=4\), day 3: \(4\times6\times\frac{4}{6}=16\), day 4: \(16\times6\times\frac{4}{6}=64\). Wait, but that would be \(a_n = 4\times a_{n - 1}\)? But let's check the initial terms. Wait, the problem says "the number of people who buy the song on day \(n\)". So \(a_1\) is the number on day 1, which is 1. Then \(a_2\) is 4 (4 out of 6 friends of Jaime bought it). Then \(a_3\): each of the 4 people on day 2 played it for 6 friends, so 4×6 friends, and 4 out of 6 of those buy it, so \(a_3=4\times\frac{4}{6}\times6 = 4\times4=16\)? Wait, the \(\frac{4}{6}\) and 6 cancel out, so \(a_3 = 4\times a_2\). So the recursive formula is \(a_n=a_{n - 1}\times4\)? But wait, the numbers given: we have \(\frac{4}{6}\), \(a_{n-1}\), 4, 6, etc. Wait, maybe I made a mistake. Wait, the problem says "Drag the needed numbers and operations". Let's re - examine.

Wait, \(a_1\) is the number of people who buy on day 1, which is 1. So \(a_1 = 1\). Then for \(a_n\), the number of people who buy on day \(n\): on day \(n - 1\), \(a_{n - 1}\) people bought it. Each of those \(a_{n - 1}\) people played it for 6 friends, so there are \(a_{n - 1}\times6\) friends. Then 4 out of 6 of those friends buy the song, so the number of buyers on day \(n\) is \(a_n=a_{n - 1}\times6\times\frac{4}{6}=a_{n - 1}\times4\). Wait, but the available numbers are 0, \(\frac{4}{6}\), 1, 2, 4, 6, 8, \(a_{n-1}\), \(a_{n + 1}\), +, ×, etc. Wait, maybe the formula is \(a_n=a_{n - 1}\times\frac{4}{6}\times6\), but \(\frac{4}{6}\times6 = 4\), so \(a_n=a_{n - 1}\times4\). But let's check with the initial terms. \(a_1 = 1\), \(a_2=1\times4 = 4\), \(a_3=4\times4 = 16\), which matches the earlier calculation. So \(a_1 = 1\), and \(a_n=a_{n - 1}\times4\)? But the available numbers: we have \(\frac{4}{6}\), \(a_{n-1}\), 4, 6, etc. Wait, maybe the correct recursive formula is \(a_n=a_{n - 1}\times\frac{4}{6}\times6\), but \(\frac{4}{6}\times6 = 4\), so \(a_n=a_{n - 1}\times4\). But let's see the options. The first blank is \(a_1\), which is 1. The second blank: \(a_n=a_{n - 1}\times4\)? But the available numbers include \(\frac{4}{6}\), 4, \(a_{n-1}\), 6, etc. Wait, maybe I messed up. Wait, the number of people who buy on day \(n\): on day \(n-1\), \(a_{n - 1}\) people bought it. Each of those \(a_{n - 1}\) people played it for 6 friends, so total friends: \(a_{n - 1}\times6\). Then 4 out of 6 of those friends buy it, so \(a_n=(a_{n - 1}\times6)\times\frac{4}{6}=a_{n - 1}\times4\). So \(a_1 = 1\), and \(a_n=a_{n - 1}\times4\). But the available numbers for \(a_1\) is 1, and for \(a_n\) is \(a_{n - 1}\times4\) (using \(a_{n-1}\) and 4 with multiplication).

Answer:

\(a_1=\boldsymbol{1}\)

\(a_n=\boldsymbol{a_{n - 1}\times4}\) (or using the available numbers, since \(\frac{4}{6}\times6 = 4\), but the simplest is \(a_n=a_{n - 1}\times4\) with \(a_1 = 1\))