Questions: If a5=17 and a12=4561, find the common ratio for the sequence. Round your answer to the nearest hundredth, if necessary.

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.
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Solution

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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.

Final Answer

\(\boxed{2.50}\)

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