Question

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

Answer 1

Answer:

Option B is correct

$a_1=759,375;\text{ }a_n=a_{n-1}\cdot\frac{1}{15}$

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"

$\begin{gathered} \frac{ar^5}{ar^4}=\frac{1}{15} \\ r=\frac{1}{15} \end{gathered}$

The standard recursive function is given as:

$a_n=r\cdot a_{n-1}$

Substitute the value for the common ratio to have;

$a_n=\frac{1}{15}\cdot a_{n-1}$

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

$\begin{gathered} \\ a_n=ar^{n-1} \\ a_5=ar^4 \\ 15=a(\frac{1}{15})^4 \\ 15=a(\frac{1}{50,625}) \\ a=15\times50,625 \\ a=759,375 \end{gathered}$

Hence the first term of the sequence is 759,375