# How do you write a rule for the nth term of the geometric term given the two terms #a_4=-8/9, a_7=-64/243#?

##### 1 Answer

# a_n = -2^(n-1)/3^(n-2)#

#### Explanation:

Suppose using standard notation for a GP sequence that the first term is

# ( a, ar, ar^2, ar^3, ar^4 , ... } #

Assuming that the first term is

# a_1 = a #

# a_2 = ar #

# a_3 = ar^2 #

#vdots#

# a_n = ar^(n-1) #

Then

And,

# \ \ (ar^6)/(ar^3) = (-64/243)/(-8/9) #

# :. r^3 = 64/243*9/8 #

# :. r^3 = 8/27 #

# :. \ \ r = 2/3 #

Subs

# a*8/27 = -8/9 #

# :. a = -8/9*27/8 #

# \ \ \ \ \ \ \ = -3 #

And so the terms form a GP with

So the

# a_n = ar^(n-1) #

# \ \ \ \ = -3(2/3)^(n-1)#

# \ \ \ \ = -2^(n-1)/3^(n-2)#

**Check**

# n=4 => a_4 = -2^3/3^2 = -8/9#

# n=7 => a_7 = -2^6/3^5 = -64/243#