Asked by Karina Rocha on May 30, 2024
Verified
Write the first five terms of the geometric sequence, given a1=−9a_{1}=-9a1=−9 and r=−13r=-\frac{1}{3}r=−31 .
A) −9,3,−1,13,−19-9,3,-1, \frac{1}{3},-\frac{1}{9}−9,3,−1,31,−91
B) −9,−3,1,−13,−19-9,-3,1,-\frac{1}{3},-\frac{1}{9}−9,−3,1,−31,−91
C) −9,27,−1,19,−127-9,27,-1, \frac{1}{9},-\frac{1}{27}−9,27,−1,91,−271
D) −27,9,1,19,−127-27,9,1, \frac{1}{9},-\frac{1}{27}−27,9,1,91,−271
E) −27,9,−1,−13,−127- 27,9 , - 1 , - \frac { 1 } { 3 } , - \frac { 1 } { 27 }−27,9,−1,−31,−271
Geometric Sequence
An array of numbers wherein each element, after the first, is calculated by multiplying the previous element by a stable, non-zero quantity referred to as the common ratio.
First Five Terms
Denotes the initial five elements or numbers in a sequence or series.
- Diagnose and classify sequences as arithmetic or geometric.
- Evaluate the uniform ratio prevalent in geometric sequences.
Verified Answer
ZK
Zybrea KnightJun 04, 2024
Final Answer :
A
Explanation :
The formula for the $n$th term of a geometric sequence is $a_{n}=a_{1}r^{n-1}$. Plugging in $a_{1}=-9$ and $r=-\frac{1}{3}$, we get $a_{n}=-9\cdot\left(-\frac{1}{3}\right)^{n-1}$. Thus, the first five terms are:
a1=−9a2=−9⋅(−13)1=3a3=−9⋅(−13)2=−1a4=−9⋅(−13)3=13a5=−9⋅(−13)4=−19\begin{align*}a_{1}&=-9\\a_{2}&=-9\cdot\left(-\frac{1}{3}\right)^{1}=\boxed{3}\\a_{3}&=-9\cdot\left(-\frac{1}{3}\right)^{2}=\boxed{-1}\\a_{4}&=-9\cdot\left(-\frac{1}{3}\right)^{3}=\boxed{\frac{1}{3}}\\a_{5}&=-9\cdot\left(-\frac{1}{3}\right)^{4}=\boxed{-\frac{1}{9}}\end{align*}a1a2a3a4a5=−9=−9⋅(−31)1=3=−9⋅(−31)2=−1=−9⋅(−31)3=31=−9⋅(−31)4=−91
Therefore, the answer is $\boxed{\textbf{(A)}}.$
a1=−9a2=−9⋅(−13)1=3a3=−9⋅(−13)2=−1a4=−9⋅(−13)3=13a5=−9⋅(−13)4=−19\begin{align*}a_{1}&=-9\\a_{2}&=-9\cdot\left(-\frac{1}{3}\right)^{1}=\boxed{3}\\a_{3}&=-9\cdot\left(-\frac{1}{3}\right)^{2}=\boxed{-1}\\a_{4}&=-9\cdot\left(-\frac{1}{3}\right)^{3}=\boxed{\frac{1}{3}}\\a_{5}&=-9\cdot\left(-\frac{1}{3}\right)^{4}=\boxed{-\frac{1}{9}}\end{align*}a1a2a3a4a5=−9=−9⋅(−31)1=3=−9⋅(−31)2=−1=−9⋅(−31)3=31=−9⋅(−31)4=−91
Therefore, the answer is $\boxed{\textbf{(A)}}.$
Learning Objectives
- Diagnose and classify sequences as arithmetic or geometric.
- Evaluate the uniform ratio prevalent in geometric sequences.
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