Questions: Find the common ratio and write out the first four terms of the geometric sequence (2^(n+3))/3 Common ratio is a1= a2= a3= a4=

Solution Steps

To solve the problem, we need to identify the common ratio of the given geometric sequence and then calculate the first four terms. The general form of the sequence is given by \( \frac{2^{n+3}}{3} \). We can find the common ratio by dividing the second term by the first term. Then, we can generate the first four terms by substituting \( n = 1, 2, 3, 4 \) into the sequence formula.

The common ratio \( r \) of the geometric sequence can be calculated by dividing the second term \( a_2 \) by the first term \( a_1 \): \[ r = \frac{a_2}{a_1} = \frac{10.6667}{5.3333} = 2.0 \]

The first four terms of the sequence are calculated as follows:

• For \( n = 1 \): \[ a_1 = \frac{2^{1+3}}{3} = \frac{2^4}{3} = \frac{16}{3} \approx 5.3333 \]
• For \( n = 2 \): \[ a_2 = \frac{2^{2+3}}{3} = \frac{2^5}{3} = \frac{32}{3} \approx 10.6667 \]
• For \( n = 3 \): \[ a_3 = \frac{2^{3+3}}{3} = \frac{2^6}{3} = \frac{64}{3} \approx 21.3333 \]
• For \( n = 4 \): \[ a_4 = \frac{2^{4+3}}{3} = \frac{2^7}{3} = \frac{128}{3} \approx 42.6667 \]

Final Answer