Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term. -10, 30, -90, ... Write your answer usi... Write an equation to describe the sequence below. Use n to represent the position of a term in the sequence, where n = 1 for the first term. -10, 30, -90, ... Write your answer using decimals and integers.
Understand the Problem
The question is asking us to write an equation that describes a given geometric sequence. We need to identify the common ratio and express the nth term of the sequence using the variable n, where n represents the position of a term in the sequence.
Answer
$$ a_n = -10 \cdot (-3)^{n - 1} $$
Answer for screen readers
$$ a_n = -10 \cdot (-3)^{n - 1} $$
Steps to Solve
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Identify the first term and common ratio The first term of the sequence is $a_1 = -10$. To find the common ratio $r$, we divide the second term by the first term: $$ r = \frac{30}{-10} = -3 $$
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Write the general formula for a geometric sequence The nth term of a geometric sequence can be expressed using the formula: $$ a_n = a_1 \cdot r^{n - 1} $$
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Substitute the values into the formula Now, we substitute the first term and common ratio into the formula: $$ a_n = -10 \cdot (-3)^{n - 1} $$
$$ a_n = -10 \cdot (-3)^{n - 1} $$
More Information
The formula $a_n = -10 \cdot (-3)^{n - 1}$ describes the nth term of the sequence, where each term is derived by multiplying the previous term by the common ratio $-3$. This results in alternating signs and increasing absolute values.
Tips
- Mixing up the first term and common ratio; make sure to clearly identify them correctly.
- Forgetting to use the correct power for $r$; it's always $n - 1$ for the nth term.
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