Certainly! Let's determine the formula for the nth term of the given sequence: 3, 9, 27, 81...
1. Identify the Sequence Type:
- Look at the ratios of consecutive terms.
- 9 ÷ 3 = 3
- 27 ÷ 9 = 3
- 81 ÷ 27 = 3
- Since each term is obtained by multiplying the previous term by 3, this is a geometric sequence.
2. Find the First Term (a) and the Common Ratio (r):
- The first term (a) is 3.
- The common ratio (r) is 3.
3. General Formula for the nth Term of a Geometric Sequence:
- The nth term of a geometric sequence can be written as:
[tex]\[
a_n = a \cdot r^{(n-1)}
\][/tex]
- Here, [tex]\( a = 3 \)[/tex] and [tex]\( r = 3 \)[/tex].
4. Substitute the Values:
- Substitute the values of [tex]\( a \)[/tex] and [tex]\( r \)[/tex] into the general formula:
[tex]\[
a_n = 3 \cdot 3^{(n-1)}
\][/tex]
5. Simplify the Expression:
- Simplify the expression using the property of exponents:
[tex]\[
a_n = 3^{(1 + (n-1))} = 3^n
\][/tex]
So, the expression for the nth term of the sequence 3, 9, 27, 81,... is:
[tex]\[
a_n = 3^n
\][/tex]
This formula will give you the nth term of the sequence for any positive integer [tex]\( n \)[/tex] starting from 1.
