How do you find the explicit rule for the geometric sequence #15, 3, 3/5, 3/25#?

1 Answer
J'Neal
Apr 10, 2017

By finding the common ratio(r) of the sequence and using the formula #a_n=a_1r^(n-1)#

The common ratio is #1/5#, and the formula for this sequence is is #15/5^(n-1)#

Explanation:

The ratio from 15 to 3 is 5:1; 15 is 5 times greater than 3. This is the same from 3 to #3/5# and #3/5# to #3/25#. This gives us a common ratio of #1/5#. With this (and #a_1# being 15), we plug into the formula and get:

#a_n=15 · (1/5)^(n-1)=15 · (1/5^(n-1))=15/5^(n-1)#

To check this, we plug each number into the sequence (starting with the number 1):

  1. #15/5^((1)-1)# = #15/5^(0)# = #15/1# = 15

  2. #15/5^((2)-1)# = #15/5^(1)# = #15/5# = 3

  3. #15/5^((3)-1)# = #15/5^(2)# = #15/25# = #3/5#

  4. #15/5^((4)-1)# = #15/5^(3)# = #15/125# = #3/25#

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