To find the [tex]\( 13^{\text{th}} \)[/tex] term of the sequence [tex]\( 10, 30, 90, 270, \ldots \)[/tex], we need to identify the type of the sequence and then use the appropriate formula.
1. Determine the type of sequence:
The given sequence is a geometric sequence. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio [tex]\( r \)[/tex].
2. Identify the first term and the common ratio:
- The first term ([tex]\( a \)[/tex]) is 10.
- The next term (30) is obtained by multiplying the first term by 3 (since [tex]\( 10 \times 3 = 30 \)[/tex]).
- Similarly, [tex]\( 30 \times 3 = 90 \)[/tex] and [tex]\( 90 \times 3 = 270 \)[/tex].
Thus, the common ratio [tex]\( r \)[/tex] is 3.
3. 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 given by:
[tex]\[
a_n = a \cdot r^{(n-1)}
\][/tex]
where [tex]\( a \)[/tex] is the first term, [tex]\( r \)[/tex] is the common ratio, and [tex]\( n \)[/tex] is the term number.
4. Substitute the known values into the formula:
- First term, [tex]\( a = 10 \)[/tex]
- Common ratio, [tex]\( r = 3 \)[/tex]
- We want the [tex]\( 13^{\text{th}} \)[/tex] term, so [tex]\( n = 13 \)[/tex]
5. Calculate the [tex]\( 13^{\text{th}} \)[/tex] term:
[tex]\[
a_{13} = 10 \cdot 3^{(13-1)} = 10 \cdot 3^{12}
\][/tex]
By calculating this, we get:
[tex]\[
a_{13} = 5314410
\][/tex]
Therefore, the [tex]\( 13^{\text{th}} \)[/tex] term of the sequence is [tex]\( 5314410 \)[/tex].
