Question

Find a_1 for the geometric sequence with the given terms. a_3 = 54 and a_5 = 486

Answer 1

ANSWER

$6$

EXPLANATION

We want to find the first term of the sequence.

The general equation for the nth term a geometric sequence is written as:

$a_n=ar^{n-1}$

where a = first term; r = common ratio

Let us use this to write the equations for the third term and the fifth term.

For the third term, n = 3:

$\begin{gathered} a_3=ar^2 \\ \Rightarrow54=ar^2 \end{gathered}$

For the fifth term, n = 5:

$\begin{gathered} a_5=ar^4 \\ \Rightarrow486=ar^4 \end{gathered}$

Let us make a the subject of both formula:

$\begin{gathered} 54=ar^2_{} \\ \Rightarrow a=\frac{54}{r^2} \end{gathered}$

and:

$\begin{gathered} 486_{}=ar^4 \\ a=\frac{486}{r^4} \end{gathered}$

Now, equate both equations above and solve for r:

$\begin{gathered} \frac{54}{r^2}=\frac{486}{r^4} \\ \Rightarrow\frac{r^4}{r^2}=\frac{486}{54} \\ \Rightarrow r^{4-2}=9 \\ \Rightarrow r^2=9 \\ \Rightarrow r=\sqrt[]{9} \\ r=3 \end{gathered}$

Now that we have the common ratio, we can solve for a using the first equation for a:

$\begin{gathered} a=\frac{54}{r^2} \\ \Rightarrow a=\frac{54}{3^2}=\frac{54}{9} \\ a=6 \end{gathered}$

That is the first term.