Question

what is the next term of the geometric sequence? \\(\frac{4}{9}, -\frac{4}{3}, 4, \boxed{}\\) show calculator related content 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 and divide it by the first term:
\( r=\frac{-\frac{4}{3}}{\frac{4}{9}} \)
To divide fractions, we multiply by the reciprocal: \( r = -\frac{4}{3}\times\frac{9}{4}=-3 \)

Step2: Find the next term

The next term (fourth term) is found by multiplying the third term by the common ratio \( r \). The third term is \( 4 \), so:
\( a_4 = 4\times r = 4\times(-3) \)
\( a_4=-12 \)

Answer:

\(-12\)