Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=0 ; \quad r=\frac{1}{\pi} $$

Short answer

The fifth term is 0, and the nth term is 0.

Step-by-step solution

- Understand the formula for the nth term of a geometric sequence

The nth term of a geometric sequence, denoted as \(a_n\), is given by the formula: \[ a_n = a_1 \times r^{(n-1)} \] where \(a_1\) is the first term and \(r\) is the common ratio.

- Find the fifth term \(a_5\)

Using the formula for the nth term, substitute \(a_1 = 0\), \(r = \frac{1}{\pi}\), and \(n = 5\): \[ a_5 = 0 \times \left( \frac{1}{\pi} \right)^{4} = 0 \] Therefore, the fifth term \(a_5\) is 0.

- Find the nth term \(a_n\)

Using the formula for the nth term: \[ a_n = 0 \times \left( \frac{1}{\pi} \right)^{(n-1)} = 0 \] Thus, the nth term \(a_n\) is always 0.

Key concepts

nth term formula

The fundamental aspect of a geometric sequence is understanding how to find any term in the sequence. The formula for the nth term, denoted as \(a_n\), is: \[ a_n = a_1 \times r^{(n-1)} \] Here, \(a_1\) is the first term of the sequence, and \(r\) is the common ratio. This formula enables us to find the value of any term if we know the starting value and the common ratio.
For example, in the given exercise, the first term \(a_1\) is 0 and the common ratio \(r\) is \(\frac{1}{\text{pi}}\). By substituting these values into the formula, it becomes clear that: \[ a_n = 0 \times \frac{1}{\text{pi}}^{(n-1)} = 0 \] Therefore, every term in this sequence remains 0, due to the initial term \(a_1\) being zero.

common ratio

In a geometric sequence, the common ratio \(r\) is the factor that each term is multiplied by to get the next term. It plays a crucial role in defining the sequence's behavior. The common ratio is found by dividing any term by the previous term in the sequence.
Mathematically, if \(a_1\), \(a_2\), \(a_3\), ... are terms of the sequence, the common ratio \(r\) can be written as: \[ r = \frac{a_n}{a_{n-1}} \] In the provided exercise, the common ratio is given as \(r = \frac{1}{\text{pi}}\). This implies that each term in the sequence is obtained by multiplying the previous term by \(\frac{1}{\text{pi}}\). However, since the first term \(a_1\) is 0, all subsequent terms also remain 0, no matter the value of \(r\).

sequence term calculation

To calculate any term in a geometric sequence, you can use the nth term formula. This requires knowing the first term and the common ratio. The calculation involves raising the common ratio to the power of \((n-1)\) and then multiplying it by the first term.
For instance, to find the 5th term (\(a_5\)) in our sequence: \[ a_5 = 0 \times \frac{1}{\text{pi}}^{4} = 0 \] Even though we might be multiplying by different powers of the common ratio, the presence of the zero term always results in zero for any term we calculate in this example.
Understanding these steps is essential for mastering geometric sequences. Whether the terms are zero or another number, the process is the same and provides a structured method to determine any term within a geometric sequence.