Question

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

Answer 1

`6`

Step-by-step 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=(54)/(r^2) \end{gathered}`

and:

`\begin{gathered} 486_{}=ar^4 \\ a=(486)/(r^4) \end{gathered}`

Now, equate both equations above and solve for r:

`\begin{gathered} (54)/(r^2)=(486)/(r^4) \\ \Rightarrow(r^4)/(r^2)=(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=(54)/(r^2) \\ \Rightarrow a=(54)/(3^2)=(54)/(9) \\ a=6 \end{gathered}`

That is the first term.