How do you write the next 4 terms in each pattern and write the pattern rule given 12, 36, 84, 180, 372?

1 Answer
Nov 21, 2016

Answer:

The next #4# terms are: #756#, #1524#, #3060#, #6132#

A recursive rule is:

#{ (a_1 = 12), (a_(n+1) = 2a_n+12) :}#

A formula for the general term is:

#a_n = 3*2^(n+2)-12#

Explanation:

We can write a recursive rule for the sequence as follows:

#{ (a_1 = 12), (a_(n+1) = 2a_n+12) :}#

Dividing the sequence by #12# we get the sequence:

#1, 3, 7, 15, 31#

Compare this with the geometric sequence:

#2, 4, 8, 16, 32#

Notice that the terms of the original sequence divided by #12# are just #1# less than the terms of this sequence.

Hence we can write a general formula:

#a_n = (2^n-1)*12 = 3*2^(n+2)-12#

Use the recursive formula to find:

#a_6 = 2a_5+12 = 2*372+12 = 756#

#a_7 = 2a_6+12 = 2*756+12 = 1524#

#a_8 = 2a_7+12 = 2*1524+12 = 3060#

#a_9 = 2a_8+12 = 2*3060+12 = 60132#