To find the explicit formula for the given sequence: 33, 11, 11/3, 11/9, 11/27, ...
1. Identify the type of sequence:
- Upon inspection, this sequence is geometric because each term is obtained by multiplying the previous term by a constant factor.
2. Determine the first term:
- The first term [tex]\( a_1 \)[/tex] is 33.
3. Find the common ratio:
- The common ratio [tex]\( r \)[/tex] can be found by dividing the second term by the first term.
[tex]\[
r = \frac{11}{33} = \frac{1}{3}
\][/tex]
- To confirm, check the ratio between other consecutive terms:
[tex]\[
\frac{\frac{11}{3}}{11} = \frac{11/3}{11} = \frac{1}{3}, \quad \frac{\frac{11}{9}}{\frac{11}{3}} = \frac{11/9}{11/3} = \frac{1}{3}
\][/tex]
4. Write the general formula for the [tex]\(n\)[/tex]th term of a geometric sequence:
- The general formula for the [tex]\(n\)[/tex]th term of a geometric sequence is
[tex]\[
a_n = a_1 \cdot r^{n-1}
\][/tex]
- Substituting the values of [tex]\(a_1\)[/tex] and [tex]\(r\)[/tex]:
[tex]\[
a_n = 33 \cdot \left( \frac{1}{3} \right)^{n-1}
\][/tex]
This gives us the explicit formula for the sequence:
[tex]\[
a_n = 33 \cdot \left( \frac{1}{3} \right)^{n-1}
\][/tex]
