The explicit rule for the geometric sequence is [tex]a_n=218.75(0.8)^{n-1}[/tex]
Explanation:
The 3rd term of the sequence is 140.
The 5th term of the sequence is 89.6.
The formula for explicit rule of the geometric sequence is given by
[tex]a_n=a_1r^{n-1}[/tex]
Now, we shall find the values of [tex]a_1[/tex] and [tex]r[/tex]
Substituting the values of 3rd term and 5th term of the sequence in the explicit formula, we get,
[tex]140=a_1r^{3-1}\implies140=a_1r^{2}[/tex] -------------(1)
[tex]89.6=a_1r^{5-1}\implies89.6=a_1r^{4}[/tex] ------------(2)
Dividing (2) by (1) , we have,
[tex]\frac{89.6}{140} =\frac{a_1r^4}{a_1r^2}[/tex]
Simplifying, we get,
[tex]0.64=r^2[/tex]
Taking square root on both sides,
[tex]r=0.8[/tex]
Substituting [tex]r=0.8[/tex] in equation (1), we have,
[tex]140=a_1(0.8)^{2}\\140=a_1(0.64)\\218.75=a_1[/tex]
Thus, the values of [tex]a_1[/tex] and [tex]r[/tex] are [tex]218.75[/tex] and [tex]0.8[/tex]
Substituting these values in the explicit formula, we get,
[tex]a_n=218.75(0.8)^{n-1}[/tex]
Thus, the explicit rule for the geometric sequence is [tex]a_n=218.75(0.8)^{n-1}[/tex]
