Question

11. Determine if the following sequence is arithmetic or geometric. Then, find the 12th term. 2, 6, 18, 54, ... a. arithmetic: 35 b. arithmetic: 354,294 c. geometric: 35 d. geometric: 354,294

Answer 1

We have the sequence: 2, 6, 18, 54...

If the sequence is arithmetic, there must be a common difference between the terms that remains constant.

This is not the case for this sequence.

We can try by seeing if there is a common factor k such that:

`a_n=k\cdot a_(n-1)`

We can do it by:

`(a_2)/(a_1)=(6)/(2)=3`
`(a_3)/(a_2)=(18)/(6)=3`
`(a_4)/(a_3)=(54)/(18)=3`

There, we have a geometric sequence, with factor k=3:

`a_n=3\cdot a_(n-1)`

We can relate it to the first term as:

`\begin{gathered} a_2=3\cdot a_1 \\ a_3=3\cdot a_2=3\cdot3\cdot a_1=3^2\cdot a_1 \\ a_4=3\cdot a_3=3\cdot3^2\cdot a_1=3^3\cdot a_1 \\ a_n=3^(n-1)\cdot a_1=3^(n-1)_{}\cdot2 \end{gathered}`

For n=12, we have:

`a_(12)=3^(12-1)\cdot2=3^(11)\cdot2=177,147\cdot2=354,294`

The value of a12 is 354,294.

The answer is d) Geometric, 354,294.