How do you write the next 4 terms in each pattern and write the pattern rule given 8, 12, 24, 60, 168?

1 Answer
Nov 24, 2016

Answer:

The next #4# terms are:

#492, 1464, 4380, 13128#

Recursive rule is:

#{ (a_1 = 8), (a_(n+1) = 3a_n-12) :}#

General formula is:

#a_n = 2*3^(n-1)+6#

Explanation:

Given:

#8, 12, 24, 60, 168#

Notice that all of the terms are divisible by #4#, so consider the sequence formed by dividing by #4#:

#2, 3, 6, 15, 42#

Write down the sequence of differences between consecutive terms:

#1, 3, 9, 27#

Notice that these are just powers of #3#.

Hence we can deduce a recursive rule for the original sequence:

#{ (a_1 = 8), (a_(n+1) = 3a_n - 12) :}#

This sequence must also have a general formula of the form:

#a_n = A*3^(n-1) + B#

where #A, B# are constants to be determined.

Putting #n = 1, 2# to get two equations to solve, we find:

#{ (A+B = 8), (3A+B = 12) :}#

Subtracting the second equation from #3 xx# the first equation, we find:

#2B = 12#

Hence #B=6# and #A=2#

So the general term of our sequence may be written:

#a_n = 2*3^(n-1) + 6#

Using the recusive formulation, we can find the next #4# terms:

#a_6 = 3a_5-12 = 3*168-12 = 492#

#a_7 = 3a_6-12 = 3*492-12 = 1464#

#a_8 = 3a_7-12 = 3*1464-12 = 4380#

#a_9 = 3a_9-12 = 3*4380-12 = 13128#