How do you write a rule for the nth term of the geometric sequence and then find #a_5# given #a_4=1/27, r=4/3#?

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Answer 1 · 2017-07-30T03:29:20

#a_n = a_1r^(n-1)#
#a_5 =4/81#

The #n^(th)# term of a geometric sequence with first term #a_1# and common ratio #r# is given by:

#a_n = a_1r^(n-1)#

In this example, #a_4 = 1/27 and r=4/3#

#:. 1/27 = a_1 xx (4/3)^3#

#1/27 = a_1 xx 4^3/3^3#

#1/27 = a_1 xx 64/27#

#64a_1 = 27/27#

#a_1 = 1/64#

We are asked to find #a_5# using the formula for the #n^(th)# term above.

#a_5 = a_1 xx r^4#

#= 1/64 xx (4/3)^4#

#= 1/64 xx 256/81#

#= 4/81#

NB: We could have found this result more simply by using:

#a_n = a_(n-1) xx r#

#:. a_5 = a_4 xx r#

#a_5 = 1/27 xx 4/3 = 4/81#