Question
The first term in a geometric sequence is \( \frac{1}{9} \) and the common ratio is -3. Find the \( 7^{\text {th }} \) term in the sequence. A. -243 B. \( -\frac{1}{2187} \) C. \( \frac{1}{729} \) D. 81
Ask by Floyd Floyd. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The 7th term is 81.
Solution
Given that the first term is \( a_1 = \frac{1}{9} \) and the common ratio is \( r = -3 \), the \( n^\text{th} \) term of a geometric sequence is given by
\[
a_n = a_1 \cdot r^{n-1}
\]
For the \( 7^{\text{th}} \) term, we set \( n = 7 \):
\[
a_7 = \frac{1}{9} \cdot (-3)^{7-1} = \frac{1}{9} \cdot (-3)^6
\]
Since \( (-3)^6 \) is raised to an even power, it becomes positive:
\[
(-3)^6 = 3^6 = 729
\]
Thus,
\[
a_7 = \frac{1}{9} \cdot 729 = \frac{729}{9} = 81
\]
The correct answer is 81.
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Simplify this solution Bonus Knowledge
To find the \( 7^{\text{th}} \) term in a geometric sequence, you can use the formula for the \( n^{\text{th}} \) term, which is given by: \[ a_n = a_1 \cdot r^{(n-1)} \] where: - \( a_1 \) is the first term, - \( r \) is the common ratio, - \( n \) is the term number. Given \( a_1 = \frac{1}{9} \), \( r = -3 \), and \( n = 7 \), we can substitute these values into the formula: \[ a_7 = \frac{1}{9} \cdot (-3)^{6} \] Calculating \( (-3)^6 \): \[ (-3)^6 = 729 \] Now, substitute back: \[ a_7 = \frac{1}{9} \cdot 729 = \frac{729}{9} = 81 \] So, the \( 7^{\text{th}} \) term in the sequence is 81. The correct answer is D. 81.
