Hello! To find the common ratio of a geometric sequence, you can use the formula:
\[ a_n = a_1 \times r^{(n-1)} \]
where \( a_n \) represents the \( n \)th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence.
From the given information, we know that \( a_2 = 2 \) and \( a_5 = \frac{1}{32} \). Using the formula for the \( 2 \)nd and \( 5 \)th terms, we can set up two equations:
\[ a_2 = a_1 \times r^{(2-1)} = 2 \]
\[ a_5 = a_1 \times r^{(5-1)} = \frac{1}{32} \]
Now, divide the second equation by the first equation to eliminate \( a_1 \):
\[ \frac{a_5}{a_2} = \frac{\frac{1}{32}}{2} \]
\[ \frac{1}{64} = \frac{1}{64} \times r^4 \]
From this, it's clear that \( r^4 = 1 \), which means \( r = 1 \) or \( r = -1 \) since any number raised to the power of 4 and equaling 1 has these two possible solutions.
However, since the sequence has positive terms (given \( a_2 = 2 \)), the common ratio \( r \) must be \( 1 \).
Therefore, the common ratio of the sequence is \( r = 1 \).
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Answer:
Hello! To find the common ratio of a geometric sequence, you can use the formula:
\[ a_n = a_1 \times r^{(n-1)} \]
where \( a_n \) represents the \( n \)th term, \( a_1 \) is the first term, \( r \) is the common ratio, and \( n \) is the position of the term in the sequence.
From the given information, we know that \( a_2 = 2 \) and \( a_5 = \frac{1}{32} \). Using the formula for the \( 2 \)nd and \( 5 \)th terms, we can set up two equations:
\[ a_2 = a_1 \times r^{(2-1)} = 2 \]
\[ a_5 = a_1 \times r^{(5-1)} = \frac{1}{32} \]
Now, divide the second equation by the first equation to eliminate \( a_1 \):
\[ \frac{a_5}{a_2} = \frac{\frac{1}{32}}{2} \]
\[ \frac{1}{64} = \frac{1}{64} \times r^4 \]
From this, it's clear that \( r^4 = 1 \), which means \( r = 1 \) or \( r = -1 \) since any number raised to the power of 4 and equaling 1 has these two possible solutions.
However, since the sequence has positive terms (given \( a_2 = 2 \)), the common ratio \( r \) must be \( 1 \).
Therefore, the common ratio of the sequence is \( r = 1 \).