### Problem 1: Finding the Fifth Term in the Geometric Sequence
First, let's determine the common ratio of the given geometric sequence. The first term is 9, and the second term is 18.
The common ratio [tex]\( r \)[/tex] is calculated as:
[tex]\[ r = \frac{\text{second term}}{\text{first term}} = \frac{18}{9} = 2 \][/tex]
Now, we need to find the fifth term of the geometric sequence. The general formula for the nth term [tex]\( a_n \)[/tex] of a geometric sequence is:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
Where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( r \)[/tex] is the common ratio,
- [tex]\( n \)[/tex] is the term number.
In this case, the first term [tex]\( a \)[/tex] is 9, the common ratio [tex]\( r \)[/tex] is 2, and we need the fifth term ([tex]\( n = 5 \)[/tex]).
So:
[tex]\[ a_5 = 9 \cdot 2^{5-1} \][/tex]
[tex]\[ a_5 = 9 \cdot 2^4 \][/tex]
[tex]\[ a_5 = 9 \cdot 16 \][/tex]
[tex]\[ a_5 = 144 \][/tex]
Therefore, the fifth term of the geometric sequence is 144.
### Problem 2: Finding the First Term in the Geometric Sequence
Next, let's use the information given to find the first term of another geometric sequence where the common ratio [tex]\( r \)[/tex] is 2, and the 8th term is 256.
Again, using the formula for the nth term [tex]\( a_n \)[/tex]:
[tex]\[ a_n = a \cdot r^{n-1} \][/tex]
We know:
- [tex]\( a_8 = 256 \)[/tex],
- [tex]\( r = 2 \)[/tex],
- [tex]\( n = 8 \)[/tex].
Substituting these values into the formula, we have:
[tex]\[ 256 = a \cdot 2^{8-1} \][/tex]
[tex]\[ 256 = a \cdot 2^7 \][/tex]
[tex]\[ 256 = a \cdot 128 \][/tex]
Solving for [tex]\( a \)[/tex]:
[tex]\[ a = \frac{256}{128} \][/tex]
[tex]\[ a = 2 \][/tex]
Therefore, the first term of this geometric sequence is 2.
