How do you write the next 4 terms in each pattern and write the pattern rule given 7, 25, 79, 241, 727?

1 Answer
Nov 21, 2016

Answer:

The next #4# terms are: #2185#, #6559#, #19681#, #59047#

A recursive rule is:

#{ (a_1 = 7), (a_(n+1) = 3a_n + 4) :}#

The general formula can be written:

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

Explanation:

A recursive formula for the given sequence is:

#{ (a_1 = 7), (a_(n+1) = 3a_n + 4) :}#

That is: The first term is #7# and each subsequent term is formed by multiplying by #3# then adding #4#.

Consider the sequence of powers of #3#:

#1, 3, 9, 27, 81, 243, 729,...#

Notice that if we discard the first two terms and subtract #2# from each of the resulting numbers then we get the given sequence.

So we can write a formula for a general term of the sequence as:

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

Using the recursive formula, we find:

#a_6 = 3a_5+4 = 3*727+4 = 2185#

#a_7 = 3a_6+4 = 3*2185+4 = 6559#

#a_8 = 3a_7+4 = 3*6559+4 = 19681#

#a_9 = 3a_8+4 = 3*19681+4 = 59047#