Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=2 ; \quad r=3 $$

Short answer

The fifth term is 162 and the nth term is \(2 \times 3^{(n-1)}\).

Step-by-step solution

Identify the Given Values

The first term of the geometric sequence is given as \(a_1 = 2\) and the common ratio is \(r = 3\).

Apply the Formula for the nth Term

The formula for the nth term of a geometric sequence is \(a_n = a_1 \times r^{(n-1)}\).

Calculate the Fifth Term

Substitute the values into the formula to find the fifth term: \[a_5 = 2 \times 3^{(5-1)} = 2 \times 3^4 = 2 \times 81 = 162\]. The fifth term is 162.

Express the nth Term

Substitute the given values of \(a_1\) and \(r\) into the nth term formula: \[a_n = 2 \times 3^{(n-1)}\]. This is the general formula for the nth term.

Key concepts

nth term formula

In a geometric sequence, each term after the first can be found using the \(nth\) term formula. This formula helps us to find any term in the sequence without having to list all the previous terms. The \(nth\) term formula is given by: \[a_n = a_1 \times r^{(n-1)}\]. Here, \(a_n\) represents the \(nth\) term, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence.

common ratio

The common ratio is a key concept in geometric sequences. It is the factor by which we multiply the previous term to get the next term. In other words, the common ratio is constant for all the terms in the sequence.
To identify the common ratio \(r\), you can divide any term in the sequence by its preceding term. For the given exercise, the common ratio is 3.
Here's how it works:

• If the first term is 2, multiplying it by 3 gives the second term (2 \(*\) 3 = 6).
• Multiplying the second term by 3 gives the third term (6 \(*\) 3 = 18), and so on.

first term

The first term (\(a_1\)) is the starting point of a geometric sequence. It is the initial value, from which the sequence builds by multiplying by the common ratio.
In the exercise provided, the first term \(a_1\) is 2.
Here’s a quick example to show its importance:

• Suppose you have a sequence with a different first term, say 5, and the same common ratio as before (3). The terms would be quite different: 5, 15, 45, and so on, compared to 2, 6, 18, etc.

Knowing the first term helps set up the whole sequence correctly.