Question

Hello. Im trying to assist my 9th grade daughter who is autistic with her test corrections for extra credit and so we can make sure she understands the concepts before the end of the school yr exam. Its been over 20 yrs since i had Algebra 1 and Im very rusty at it. My daughter gets easily frustrated and impatient so Im trying to do some of the leg work before going over it with her. I appreciate your assistance and patience in advance.

Answer 1

A geometric sequence is a sequence of numbers where consecutive elements have a common ratio r:

`(a_(n+1))/(a_n)=r`

Then, we can find the next element of the sequence by multiplying the preceding element by the common ratio r:

`a_(n+1)=r\cdot a_n`

If the first element of the sequence is a₁, then:

`\begin{gathered} a_2=a_1\cdot r \\ a_3=a_1r^2 \\ a_4=a_1r^3 \\ \ldots \\ a_n=a_1r^(n-1) \end{gathered}`

The given information states that a₁=8 and that the common ratio is 2/3.

Then, substitute a₁=8 and r=2/3 into the expression for the n-th term:

`a_n=8\cdot\mleft((2)/(3)\mright)^(n-1)`

Therefore, the sequence can be written as a function as follows:

`f(n)=8\cdot\mleft((2)/(3)\mright)^(n-1)`