To determine the rule of the given sequence [tex]\( \{3, 6, 12, 24, \ldots\} \)[/tex], let's analyze the pattern within the sequence step-by-step:
1. Identify the sequence terms:
The initial terms of the sequence are 3, 6, 12, 24.
2. Calculate the ratio between consecutive terms:
- The ratio of the second term to the first term is [tex]\( \frac{6}{3} = 2 \)[/tex].
- The ratio of the third term to the second term is [tex]\( \frac{12}{6} = 2 \)[/tex].
- The ratio of the fourth term to the third term is [tex]\( \frac{24}{12} = 2 \)[/tex].
3: Notice the consistent pattern:
Each term is obtained by multiplying the previous term by 2.
4. Formalize the rule:
From the observed pattern, the [tex]\(n\)[/tex]-th term can be written as follows:
[tex]\[
a_n = 3 \cdot 2^{(n-1)}
\][/tex]
where [tex]\( n \)[/tex] is the position of the term in the sequence.
5. Verify the rule with the given terms:
- For [tex]\( n = 1 \)[/tex]: [tex]\( a_1 = 3 \cdot 2^{(1-1)} = 3 \cdot 2^0 = 3 \cdot 1 = 3 \)[/tex]
- For [tex]\( n = 2 \)[/tex]: [tex]\( a_2 = 3 \cdot 2^{(2-1)} = 3 \cdot 2^1 = 3 \cdot 2 = 6 \)[/tex]
- For [tex]\( n = 3 \)[/tex]: [tex]\( a_3 = 3 \cdot 2^{(3-1)} = 3 \cdot 2^2 = 3 \cdot 4 = 12 \)[/tex]
- For [tex]\( n = 4 \)[/tex]: [tex]\( a_4 = 3 \cdot 2^{(4-1)} = 3 \cdot 2^3 = 3 \cdot 8 = 24 \)[/tex]
Thus, the rule for the sequence [tex]\( \{3, 6, 12, 24, \ldots\} \)[/tex] is:
[tex]\[
a_n = 3 \cdot 2^{(n-1)}
\][/tex]
Now, let's determine the next term using the rule:
- For [tex]\( n = 5 \)[/tex]: [tex]\( a_5 = 3 \cdot 2^{(5-1)} = 3 \cdot 2^4 = 3 \cdot 16 = 48 \)[/tex]
So, the next term in the sequence is 48, which confirms that the rule applied correctly.
