Answer:
an = (5/16)2^(n -1)
Explanation:
The explicit formula for the n-th term of a geometric sequence is ...
an = a1·r^(n-1)
Filling in the given values, we get two equations in a1 and r:
a5 = 5 = a1·r^(5 -1)
a9 = 80 = a1·r(9 -1)
Dividing the second equation by the first gives ...
80/5 = (a1·r^8)/(a1·r^4)
16 = r^4
2 = r . . . . . . . . because we know 2^4 = 16. You could also solve using logarithms
Then we can find a1 from ...
5 = a1·r^4 = 16·a1
5/16 = a1
The explicit formula is ...
an = (5/16)2^(n -1)
__
The above formula matches the form of the equation usually expected. However, we can make the fraction go away by adjusting the exponent:
an = 5(2^-4)(2^(n-1))
an = 5·2^(n -5)