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.

$$\{s_n\} = \{5^n\}$$

Solution

To show that the sequence is geometric, we need to demonstrate that the ratio of consecutive terms is constant. The given sequence is \(\{s_n\} = \{5^n\}\). Calculate the first four terms of the sequence: For \(n = 1\): \[ s_1 = 5^1 = 5 \] For \(n = 2\): \[ s_2 = 5^2 = 25 \] For \(n = 3\): \[ s_3 = 5^3 = 125 \] For \(n = 4\): \[ s_4 = 5^4 = 625 \] Now, find the common ratio \(r\) by taking the ratio of the second term to the first term: \[ r = \frac{s_2}{s_1} = \frac{25}{5} = 5 \] Check the ratio for the next pair of terms: \[ r = \frac{s_3}{s_2} = \frac{125}{25} = 5 \] \[ r = \frac{s_4}{s_3} = \frac{625}{125} = 5 \] Since the ratio is constant, the sequence \(\{s_n\} = \{5^n\}\) is geometric with a common ratio of 5. The first four terms of the sequence are \(5, 25, 125, 625\).