Question

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what is the next term of the geometric sequence?
\\(\frac{81}{25}, \frac{27}{5}, 9, \square\\)
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extending geometric sequences

Explanation:

Step1: Find the common ratio \( r \)

In a geometric sequence, the common ratio \( r \) is found by dividing a term by its previous term. Let's take the second term \( \frac{27}{5} \) and divide it by the first term \( \frac{81}{25} \).
\[
r=\frac{\frac{27}{5}}{\frac{81}{25}}=\frac{27}{5}\times\frac{25}{81}=\frac{27\times25}{5\times81}=\frac{675}{405}=\frac{5}{3}
\]
We can also check with the third term \( 9 \) and the second term \( \frac{27}{5} \).
\[
r = \frac{9}{\frac{27}{5}}=9\times\frac{5}{27}=\frac{45}{27}=\frac{5}{3}
\]
So the common ratio \( r=\frac{5}{3} \).

Step2: Find the next term

To find the next term (the fourth term), we multiply the third term \( 9 \) by the common ratio \( \frac{5}{3} \).
\[
a_{4}=9\times\frac{5}{3}=\frac{45}{3} = 15
\]

Answer:

\( 15 \)