Question

1. determine the $10^{\text{th}}$ term of a geometric sequence whose first term is 2 and whose common ratios is 3.

Explanation:

Step1: Recall the formula for the nth term of a geometric sequence

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

Step2: Identify the given values

We are given that \( a_1 = 2 \), \( r = 3 \), and we need to find the 10th term, so \( n = 10 \).

Step3: Substitute the values into the formula

Substitute \( a_1 = 2 \), \( r = 3 \), and \( n = 10 \) into the formula:
\( a_{10} = 2 \times 3^{(10 - 1)} \)
\( a_{10} = 2 \times 3^9 \)

Step4: Calculate \( 3^9 \)

\( 3^9 = 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 \times 3 = 19683 \)

Step5: Calculate \( a_{10} \)

\( a_{10} = 2 \times 19683 = 39366 \)

Answer:

The 10th term of the geometric sequence is \( \boxed{39366} \).