Here, the first term a is (81)/(16) and the common ratio r is (2)/(3). The sequence represents a geometric progression with general term a_n = a r^(n-1).

Observe the sequence and find the next term: 81/16, 27/8, 9/4, ?

Answer: $\frac{3}{2}$

Step-by-step solution

Step 1: Identify the type of sequence and check consecutive ratios

To identify the pattern, we find the ratio of consecutive terms. Since the ratio of the second term to the first term equals the ratio of the third term to the second term, which is $\frac{2}{3}$ , the terms form a Geometric Progression (G.P.).

Step 2: State the first term and common ratio

Here, the first term $a$ is $\frac{81}{16}$ and the common ratio $r$ is $\frac{2}{3}$ . The sequence represents a geometric progression with general term $a_n = a r^{n-1}$ .

Step 3: Calculate the fourth term

To obtain the next term, which is the 4th term $a_4$ , we multiply the 3rd term by the common ratio $r = \frac{2}{3}$ . This gives $\frac{9}{4} \times \frac{2}{3} = \frac{18}{12} = \frac{3}{2}$ .

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