Write an equation to describe the sequence -2, 4, -8, ... Use n to represent the position of a term in the sequence, where n = 1 for the first term.
Understand the Problem
The question is asking for an equation that represents the geometric sequence given as -2, 4, -8, ... The task is to express the nth term using n as the position in the sequence where n = 1 for the first term.
Answer
The nth term of the geometric sequence is given by $a_n = -2 \cdot (-2)^{(n-1)}$.
Answer for screen readers
The equation for the nth term of the geometric sequence is: $$ a_n = -2 \cdot (-2)^{(n-1)} $$
Steps to Solve
-
Identify the common ratio To find the equation for the geometric sequence, first determine the common ratio between the terms. The sequence is -2, 4, -8. The common ratio ($r$) can be found by dividing the second term by the first term: $$ r = \frac{4}{-2} = -2 $$
-
Write the general formula for a geometric sequence In a geometric sequence, the nth term is given by the formula: $$ a_n = a_1 \cdot r^{(n-1)} $$ where $a_1$ is the first term and $r$ is the common ratio we found.
-
Substitute the values into the formula The first term ($a_1$) is -2 and the common ratio ($r$) is -2. Therefore, substituting these values into the formula gives: $$ a_n = -2 \cdot (-2)^{(n-1)} $$
The equation for the nth term of the geometric sequence is: $$ a_n = -2 \cdot (-2)^{(n-1)} $$
More Information
This equation represents the nth term of the sequence, where each term is derived by multiplying the previous term by -2, hence demonstrating the nature of a geometric sequence.
Tips
- Forgetting to account for the negative sign in the first term or the common ratio.
- Not using the correct exponent in the formula for the nth term.
AI-generated content may contain errors. Please verify critical information
