Question

what are the values of $a_1$ and $r$ of the geometric series?
$1 + 3 + 9 + 27 + 81$
$\bigcirc$ $a_1 = 1$ and $r = \frac{1}{3}$
$\bigcirc$ $a_1 = \frac{1}{3}$ and $r = 1$
$\bigcirc$ $a_1 = 1$ and $r = 3$
$\bigcirc$ $a_1 = 3$ and $r = 1$

Explanation:

Step1: Identify \(a_1\)

In a geometric series, \(a_1\) is the first term. For the series \(1 + 3 + 9 + 27 + 81\), the first term is \(1\), so \(a_1 = 1\).

Step2: Calculate the common ratio \(r\)

The common ratio \(r\) in a geometric series is found by dividing any term by its previous term. Let's take the second term divided by the first term: \(\frac{3}{1}=3\). We can check with another pair, like \(\frac{9}{3} = 3\), \(\frac{27}{9}=3\), \(\frac{81}{27}=3\). So \(r = 3\).

Answer:

\(a_1 = 1\) and \(r = 3\) (the option with the blue dot: \(a_1 = 1\) and \(r = 3\))