To find the value that Natalia should use as the common ratio in the formula, we need to determine how each term in the sequence relates to the previous term. Let's start by examining the given sequence:
[tex]\[ 8, 12, 18, 27 \][/tex]
We'll find the ratios of successive terms to see if there's a consistent pattern.
1. Calculate the ratio between the second term and the first term:
[tex]\[ \text{Ratio}_1 = \frac{12}{8} \][/tex]
Simplifying this:
[tex]\[ \text{Ratio}_1 = \frac{12 \div 4}{8 \div 4} = \frac{3}{2} \][/tex]
2. Calculate the ratio between the third term and the second term:
[tex]\[ \text{Ratio}_2 = \frac{18}{12} \][/tex]
Simplifying this:
[tex]\[ \text{Ratio}_2 = \frac{18 \div 6}{12 \div 6} = \frac{3}{2} \][/tex]
3. Calculate the ratio between the fourth term and the third term:
[tex]\[ \text{Ratio}_3 = \frac{27}{18} \][/tex]
Simplifying this:
[tex]\[ \text{Ratio}_3 = \frac{27 \div 9}{18 \div 9} = \frac{3}{2} \][/tex]
We see that the ratio between successive terms is consistently [tex]\(\frac{3}{2}\)[/tex].
Therefore, the value Natalia should use as the common ratio in the recursive formula is:
[tex]\[ \boxed{\frac{3}{2}} \][/tex]
