The first term is 4, and the common ratio -1.5.

The n-th term is 4(-1.5)^(n – 1)

If 729/16 is the n-th term, then:

729 / 16 = 4(-1.5)^(n – 1)

729 / 64 = -(3/2)^(n – 1)

729 / 64 = (3/2)^6, and therefore n = 7.

729 / 16 is the 7th term.

(b)

If the common ratio is r, then:

4/25 = (25/4) r^4

r^4 = (4/25) / (25/4) = 16 / 625

r = +/- (4/5).

If the first term is a, then the third is ar^2.

ar^2 = 25/4

a = (25/4) / r^2

a = (25/4) / (16/25)

= 25^2 / 64

= 625 / 64

The first three terms are either:

625 / 64, 125 / 16, 25 / 4

or

625 / 64, -125 / 16, 25 / 4.

Tn = a* (r)^(n-1)

729/16 = 4* (-3/2)^(n-1)

(-3/2)^(n-1) = 729/64

(-3/2)^(n-1) = (-3/2)^6

n-1 = 6

or n = 7 …ANSWER

b) T3 = a(r)^2 = 25/4 ……(i)

T7 = a(r)^6 = 4/25 …..(ii)

Dividing (ii) by (i) we get

r^4 = (4/25)^2

r^4 = (2/5)^4 ; r = 2/5 OR -2/5

Now get the first term a