Question

Find the fifth term and the nth term of the geometric sequence whose first term \(a_{1}\) and common ratio \(r\) are given. $$ a_{1}=0 ; \quad r=\frac{1}{7} $$

Short answer

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

Step-by-step solution

Identify the General Formula for Geometric Sequence

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

Plug in the Given Values

Substitute the given values of \(a_{1} = 0\) and \(r = \frac{1}{7}\) into the formula. This gives\[ a_n = 0 \times \left(\frac{1}{7}\right)^{(n-1)} \]

Simplify the Expression for the nth Term

Simplify the expression to find the nth term\[ a_{n} = 0 \]Since any number multiplied by 0 is 0, the nth term will always be 0.

Find the Fifth Term

To find the fifth term, substitute \(n = 5\) into the simplified nth term formula.\[ a_5 = 0 \]

Key concepts

nth term of geometric sequence

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a constant called the common ratio. To find the nth term of a geometric sequence, we use the formula \[ a_n = a_{1} \times r^{(n-1)} \] where \(a_{1}\) is the first term, \(r\) is the common ratio, and \(n\) is the position of the term in the sequence. For example, in our exercise, if \(a_{1}\) is the first term and \(r\) is the common ratio, the formula becomes very helpful. By plugging in the values given: \(a_{1}=0\) and \(r=\frac{1}{7}\), we can calculate any term in the sequence. For this particular sequence, regardless of the value of \(n\), the term will always be zero because multiplying any number by zero results in zero.

common ratio

The common ratio in a geometric sequence is the factor by which we multiply each term to get to the next term. It is a constant value that defines the sequence's progression. To find the common ratio \(r\), you can take any term in the sequence and divide it by the previous term. Mathematically, it is represented as: \[ r = \frac{a_{n}}{a_{n-1}} \] In our exercise, the common ratio given is \( \frac{1}{7} \). This tells us each term is \( \frac{1}{7} \) times the previous term. However, since the first term \(a_{1}\) is 0, multiplying 0 by any ratio still yields 0, demonstrating that all terms in the sequence will be 0.

sequence formula

The sequence formula for a geometric sequence helps us find any term in that sequence quickly. The standard formula used is: \[ a_n = a_{1} \times r^{(n-1)} \] Here:

• \(a_n\): nth term of the sequence
• \(a_{1}\): First term of the sequence
• \(r\): Common ratio
• \(n\): Position of the term in the sequence

By using this formula, we can determine the value of any term in the sequence. In our specific problem, the formula simplifies to \(a_n = 0\) since the first term \(a_{1}\) is 0 and any number multiplied by 0 is always 0. This formula showcases how we can derive each term efficiently in a geometric sequence. Even when terms seem trivial, it serves as a robust tool for understanding progression.