The third and sixth terms of a geometric sequence are 75 and

The third and sixth terms of a geometric sequence are -75 and -9375 respectively. Find the first term, the common ratio, and an explicit rule for the nth term. Show all work.

Solution

the nth term of a geometric series is ar^(n-1)

given third [ ar^2] = -75

and sixth term [ar^5]= -9375

now divide the above two

[ar^2] / [ar^5] = -75/-9375

1/r^3 = 1/125

so r^3 = 125

r=5

but third term a^2 = -75

a(5)^2 =-75

a(25) =-75

a = -5

so first term =-5

common ratio = 5

nth term = ar^(n-1)