The sequence of push-ups Miguel did can be described as a geometric progression.
To analyze the sequence:
- On Sunday, he did 1 push-up.
- On Monday, he did 3 push-ups.
- On Tuesday, he did 9 push-ups.
- On Wednesday, he did 27 push-ups.
To determine the pattern, we can observe the relationship between consecutive terms:
- From Sunday to Monday: \( 3 = 1 \times 3 \)
- From Monday to Tuesday: \( 9 = 3 \times 3 \)
- From Tuesday to Wednesday: \( 27 = 9 \times 3 \)
Here, each term is obtained by multiplying the previous term by 3.
The general formula for the \( n \)-th term of a geometric sequence can be expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the term number.
In this case:
- The first term \( a_1 = 1 \),
- The common ratio \( r = 3 \).
Thus, the sequence can be represented as:
- \( a_1 = 1 \)
- \( a_2 = 1 \cdot 3 = 3 \)
- \( a_3 = 1 \cdot 3^2 = 9 \)
- \( a_4 = 1 \cdot 3^3 = 27 \)
Therefore, this is a geometric sequence with a common ratio of 3.
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