Question
Lester is learning to play violin. Each month, Lester is able to play 4 times as many songs as he was able to play the month before. If you list the total number of songs that he can play each month, what kind of sequence will you see?
Answers
Answered by
GPT-4o mini
Lester's ability to play songs increases by a factor of 4 each month.
Let's denote the number of songs Lester can play in the first month as \( a_1 \). Then, the number of songs he can play in subsequent months can be represented as follows:
- In the first month, he can play \( a_1 \) songs.
- In the second month, he can play \( 4 \times a_1 \) songs.
- In the third month, he can play \( 4^2 \times a_1 \) songs.
- In the fourth month, he can play \( 4^3 \times a_1 \) songs.
Continuing this way, in the \( n \)-th month, the number of songs Lester can play is given by the formula:
\[
a_n = 4^{(n-1)} \times a_1
\]
This shows that the number of songs Lester can play each month forms a **geometric sequence**. The common ratio of this sequence is 4.
Thus, the sequence you will see for the total number of songs that he can play each month is a geometric sequence.
Let's denote the number of songs Lester can play in the first month as \( a_1 \). Then, the number of songs he can play in subsequent months can be represented as follows:
- In the first month, he can play \( a_1 \) songs.
- In the second month, he can play \( 4 \times a_1 \) songs.
- In the third month, he can play \( 4^2 \times a_1 \) songs.
- In the fourth month, he can play \( 4^3 \times a_1 \) songs.
Continuing this way, in the \( n \)-th month, the number of songs Lester can play is given by the formula:
\[
a_n = 4^{(n-1)} \times a_1
\]
This shows that the number of songs Lester can play each month forms a **geometric sequence**. The common ratio of this sequence is 4.
Thus, the sequence you will see for the total number of songs that he can play each month is a geometric sequence.