Question

Two terms in a geometric sequence are a5=15 and a6=1.What is the recursive rule that describes the sequence?

Answer 1

Option B is correct

`a_1=759,375;\text{ }a_n=a_(n-1)\cdot(1)/(15)`

Explanations:

The nth term of a geometric sequence is given as:

`a_n=ar^(n-1)`

where:

a is the first term

r is the common ratio

If two terms in a geometric sequence are a₅ = 15 and a₆ = 1, then;

`\begin{gathered} a_5=ar^(5-1) \\ ar^4=15 \end{gathered}`

Similarly;

`\begin{gathered} a_6=ar^(6-1) \\ 1=ar^5 \end{gathered}`

Take the ratio of both expressions to get the common ratio "r"

`\begin{gathered} (ar^5)/(ar^4)=(1)/(15) \\ r=(1)/(15) \end{gathered}`

The standard recursive function is given as:

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

Substitute the value for the common ratio to have;

`a_n=(1)/(15)\cdot a_(n-1)`

To get the first term a₁, we will use the equivalent nth term of the sequence above

`\begin{gathered} \\ a_n=ar^(n-1) \\ a_5=ar^4 \\ 15=a((1)/(15))^4 \\ 15=a((1)/(50,625)) \\ a=15*50,625 \\ a=759,375 \end{gathered}`

Hence the first term of the sequence is 759,375