Hall Bowers
10/14/2023 · Middle School
\( 75, \quad 1875, \ldots \) What is the common ratio for this geometric sequence In other words, what is \( r \) ?
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Step-by-step Solution
To find the common ratio \( r \) of a geometric sequence, we can use the formula:
\[
r = \frac{a_{n+1}}{a_n}
\]
where \( a_n \) is the \( n \)-th term and \( a_{n+1} \) is the next term in the sequence.
In this case, we have the first term \( a_1 = 75 \) and the second term \( a_2 = 1875 \). We can calculate the common ratio \( r \) as follows:
\[
r = \frac{a_2}{a_1} = \frac{1875}{75}
\]
Now, we can simplify this fraction:
\[
r = \frac{1875 \div 75}{75 \div 75} = \frac{25}{1} = 25
\]
Thus, the common ratio \( r \) for the geometric sequence is
\[
\boxed{25}
\]
Quick Answer
The common ratio \( r \) is 25.
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