A geometric sequence is defined by the explicit formula #a_n = 5(-3)_(n-1)#, what is the recursive formula for the nth term of this sequence?

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Answer 1 · 2016-09-07T07:56:50

The recursive formula is:

#{(a_1=5),(a_{n+1}=-3a_n):}#

I assume the formula is: #a_n=5(-3)^{n-1}#.

To calculate the recursive formula first we can calculate some terms of the sequence:

#a_1=5*(-3)^(1-1)=5*(-3)^0=5#

#a_2=5*(-3)^(2-1)=5*(-3)^1=-15#

#a_3=5*(-3)^(3-1)=5*(-3)^2=45#

From these calculations we see that the first term is #a_1=5# and each other term comes from multiplying previous one by #-3#.

This algorithm can be written as the recursive formula:

#{(a_1=5),(a_{n+1}=-3a_n):}#