Question

how do you find the 5th term in the geometric sequence 2, 6, 18, 54,...?
a. $2 \cdot 3^4$
b. $2 \cdot 6^4$
c. $2^5$
d. $2 + 4(5)$

Explanation:

Step1: Recall geometric sequence formula

The formula for the \(n\)-th term of a geometric sequence is \(a_n = a_1 \cdot r^{n - 1}\), where \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the term number.

Step2: Identify \(a_1\) and \(r\)

For the sequence \(2, 6, 18, 54, \dots\), the first term \(a_1 = 2\). The common ratio \(r\) is found by dividing a term by its previous term, e.g., \(r=\frac{6}{2}=3\).

Step3: Find the 5th term

We want the 5th term, so \(n = 5\). Substitute \(a_1 = 2\), \(r = 3\), and \(n = 5\) into the formula: \(a_5=2\cdot3^{5 - 1}=2\cdot3^{4}\).

Step4: Eliminate other options

• Option b: The ratio is 3, not 6, so \(2\cdot6^{4}\) is incorrect.
• Option c: \(2^{5}\) is not related to the geometric sequence formula here.
• Option d: The formula \(2 + 4(5)\) is for an arithmetic sequence (since it uses addition of a common difference), but this is a geometric sequence, so it's incorrect.

Answer:

a. \(2 \cdot 3^{4}\)