Question

what is the next term of the geometric sequence?
\frac{27}{16}, -\frac{9}{4}, 3, \square

Explanation:

Step1: Find the common ratio \( r \)

To find the common ratio \( r \) in a geometric sequence, we divide a term by its previous term. Let's take the second term and divide it by the first term:
\( r=\frac{-\frac{9}{4}}{\frac{27}{16}} \)
When dividing fractions, we multiply by the reciprocal:
\( r = -\frac{9}{4}\times\frac{16}{27} \)
Simplify the numerator and denominator:
\( 9\) and \(27\) have a common factor of \(9\) (\(27\div9 = 3\), \(9\div9 = 1\)); \(16\) and \(4\) have a common factor of \(4\) (\(16\div4 = 4\), \(4\div4 = 1\)). So we get:
\( r=-\frac{1\times4}{1\times3}=-\frac{4}{3} \)
We can check with the third term and the second term: \( \frac{3}{-\frac{9}{4}}=3\times(-\frac{4}{9}) = -\frac{4}{3} \), which matches.

Step2: Find the next term

To find the next term (the fourth term) in a geometric sequence, we use the formula \( a_{n}=a_{n - 1}\times r \), where \( a_{n-1} \) is the previous term and \( r \) is the common ratio. Here, the third term \( a_{3}=3 \), and the common ratio \( r = -\frac{4}{3} \). So the fourth term \( a_{4} \) is:
\( a_{4}=3\times(-\frac{4}{3}) \)
The \( 3 \) in the numerator and the \( 3 \) in the denominator cancel out, leaving:
\( a_{4}=- 4 \)

Answer:

\(-4\)