To find the fourth term in a geometric sequence, we'll use the formula for the nth term of a geometric sequence, which is given by:
[tex]\[ a_n = a_1 \cdot r^{(n-1)} \][/tex]
Here:
- [tex]\( a_1 \)[/tex] is the first term of the sequence.
- [tex]\( r \)[/tex] is the common ratio.
- [tex]\( n \)[/tex] is the term number you want to find.
Given:
- [tex]\( a_1 = 10 \)[/tex]
- [tex]\( r = 0.5 \)[/tex]
- [tex]\( n = 4 \)[/tex]
We need to find the fourth term ([tex]\( a_4 \)[/tex]).
Let's substitute the given values into the formula:
[tex]\[ a_4 = 10 \cdot (0.5)^{(4-1)} \][/tex]
Simplify the exponent:
[tex]\[ a_4 = 10 \cdot (0.5)^3 \][/tex]
Calculate [tex]\( (0.5)^3 \)[/tex]:
[tex]\[ (0.5)^3 = 0.5 \times 0.5 \times 0.5 = 0.125 \][/tex]
Now multiply by the first term ([tex]\( a_1 \)[/tex]):
[tex]\[ a_4 = 10 \cdot 0.125 \][/tex]
[tex]\[ a_4 = 1.25 \][/tex]
Therefore, the value of the fourth term in the geometric sequence is:
[tex]\[ 1.25 \][/tex]
