How do you write the first five terms of the sequence defined recursively #a_1=3, a_(k+1)=2(a_k-1)#?

1 Answer
Oct 16, 2017

3, 4, 6, 10, 18

Explanation:

When a sequence is defined recursively, the previous term is used to find the next term. Start by using #a_1# to find #a_2# (second term).

Substitute #a_1# for #a_k#.
#a_(1+1)=2(a_1-1)#
#a_2=2(a_1-1)#
Because #a_1=3#, substitute #a_1# for #3# to find #a_2#.
#a_2=2(3-1)#
#a_2=2(2)#
#a_2=4# This is your second term!

Now use #a_2# to find #a_3# just like how you used #a_1# to find #a_2#.
#a_(2+1)=2(a_2-1)#
#a_3=2(4-1)#
#a_3=2(3)#
#a_3=6# This is your third term!

Repeat these steps to find #a_4# using #a_3#.
#a_(3+1)=2(a_3-1)#
#a_4=2(6-1)#
#a_4=2(5)#
#a_4=10# This is your fourth term!

Find #a_5# using #a_4#.
#a_(4+1)=2(a_4-1)#
#a_5=2(10-1)#
#a_5=2(9)#
#a_5=18# This is your fifth term!

List the terms from least to greatest and separate the terms with commas.

Answer: 3, 4, 6, 10, 18