Questions: If a5=17 and a12=4561, find the common ratio for the sequence. Round your answer to the nearest hundredth, if necessary.
Transcript text: If $a_{5}=17$ and $a_{12}=4561$, find the common ratio for the sequence. Round your answer to the nearest hundredth, if necessary.
Solution
Solution Steps
Step 1: Identify the type of sequence
The problem provides two terms of a sequence, \( a_{5} = 17 \) and \( a_{12} = 4561 \). Since the terms are growing rapidly, this suggests the sequence is geometric, where each term is multiplied by a common ratio \( r \).
Step 2: Write the general formula for a geometric sequence
The general formula for the \( n \)-th term of a geometric sequence is:
\[
a_{n} = a_{1} \cdot r^{n-1}
\]
where \( a_{1} \) is the first term and \( r \) is the common ratio.
Step 3: Set up equations using the given terms
Using the formula, write equations for \( a_{5} \) and \( a_{12} \):
\[
a_{5} = a_{1} \cdot r^{4} = 17
\]
\[
a_{12} = a_{1} \cdot r^{11} = 4561
\]
Step 4: Divide the equations to eliminate \( a_{1} \)
Divide the second equation by the first to eliminate \( a_{1} \):
\[
\frac{a_{12}}{a_{5}} = \frac{a_{1} \cdot r^{11}}{a_{1} \cdot r^{4}} = r^{7}
\]
\[
\frac{4561}{17} = r^{7}
\]
\[
268.2941 \approx r^{7}
\]
Step 5: Solve for \( r \)
Take the 7th root of both sides to solve for \( r \):
\[
r = \sqrt[7]{268.2941} \approx 2.5
\]
Step 6: Round the answer to the nearest hundredth
The common ratio \( r \) is approximately \( 2.5 \), which is already rounded to the nearest hundredth.