Question: Show that the given sequence is geometric. Then, find the common ratio and write out the first four…

Show that the given sequence is geometric. Then, find the common ratio and write out the first four terms.

$$\{s_n\} = \{4^n\}$$

Solution

To show that the given sequence is geometric and find the common ratio, we will analyze the sequence \(\{s_n\} = \{4^n\}\). A sequence is geometric if the ratio of consecutive terms is constant. First, find the first term: \[ s_1 = 4^1 = 4 \] Find the second term: \[ s_2 = 4^2 = 16 \] Find the third term: \[ s_3 = 4^3 = 64 \] Find the fourth term: \[ s_4 = 4^4 = 256 \] Next, find the common ratio by dividing the second term by the first term: \[ \text{Common Ratio} = \frac{s_2}{s_1} = \frac{16}{4} = 4 \] Verify with the third and second term: \[ \text{Common Ratio} = \frac{s_3}{s_2} = \frac{64}{16} = 4 \] Verify with the fourth and third term: \[ \text{Common Ratio} = \frac{s_4}{s_3} = \frac{256}{64} = 4 \] The common ratio is consistent, confirming the sequence is geometric. The first four terms of the sequence are 4, 16, 64, and 256. The common ratio is 4.