Please answer this calculus question?

Question

The second and fifth terms of a geometric sequence are -4 and 1/2, respectively. Find

 

a) the first term,

b) the common ratio,

c) an explicit rule for the nth term, and

d) a recursive rule for the nth term. 

 

Please show all your work.

Answer 1

The sequence is a, ar, ar², ar³, ar⁴,...

The given data is that ar=-4 and ar⁴=1/2

divide ar⁴/ar=r³=(1/2)/-4=-1/8

r=-1/2 the common ratio

ar=-4 gives that a=-4/(-1/2)=8 the first term

sequence is 8, -4, 2, -1, 1/2, -1/4, ....

a_n=8(-1/2)ⁿ for n=0,1,2,3,....

as with every geometric sequence a_n+1=ra_n a recursive rule