Howell Delgado
05/29/2023 · Senior High School
\( 1,4,9,16, \ldots= \) general \( 100,50,25,12.5, \ldots= \) geometric \( 1,8,27,64 \ldots= \)
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Step-by-step Solution
Let's analyze the sequences you've provided:
1. The first sequence \( 1, 4, 9, 16, \ldots \) consists of perfect squares. The general term for this sequence can be expressed as:
\[
a_n = n^2
\]
where \( n = 1, 2, 3, 4, \ldots \)
2. The second sequence \( 100, 50, 25, 12.5, \ldots \) is a geometric sequence. The first term is \( 100 \) and the common ratio is \( \frac{1}{2} \). The general term for this sequence can be expressed as:
\[
a_n = 100 \left( \frac{1}{2} \right)^{n-1}
\]
where \( n = 1, 2, 3, \ldots \)
3. The third sequence \( 1, 8, 27, 64, \ldots \) consists of perfect cubes. The general term for this sequence can be expressed as:
\[
a_n = n^3
\]
where \( n = 1, 2, 3, 4, \ldots \)
In summary:
- The first sequence is \( n^2 \).
- The second sequence is \( 100 \left( \frac{1}{2} \right)^{n-1} \).
- The third sequence is \( n^3 \).
Quick Answer
- First sequence: \( n^2 \)
- Second sequence: \( 100 \left( \frac{1}{2} \right)^{n-1} \)
- Third sequence: \( n^3 \)
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