Problem 62 - Algebra 2

Question

Write the explicit formula for each geometric sequence. Then generate the first three terms. $$ a_{1}=-7, r=0.1 $$

Step-by-Step Solution

Answer

The explicit formula for the geometric sequence is \(a_n = -7 \cdot 0.1^{(n-1)}\). The first three terms of the sequence are \(-7, -0.7, -0.07\) respectively.

Step 1: Write the explicit formula for the geometric sequence

The general formula for the nth term of a geometric sequence is \(a_n = a_{1} \cdot r^{(n-1)}\). Here, \(a_{1} = -7\) is the first term and \(r = 0.1\) is the common ratio, where \(n\) is the term sequence.

Step 2: Substitute \(a_{1}\) and \(r\) into the formula

Substitute \(a_{1} = -7\) and \(r = 0.1\) into the formula to get the explicit formula: \(a_n = -7 \cdot 0.1^{(n-1)}\).

Step 3: Generate the first three terms

Substitute \(n = 1, 2, 3\) into the formula to get the first three terms: \(a_1 = -7 \cdot 0.1^{(1-1)} = -7\), \(a_2 = -7 \cdot 0.1^{(2-1)} = -0.7\), \(a_3 = -7 \cdot 0.1^{(3-1)} = -0.07\).

Key Concepts

Explicit Formula

The explicit formula is a crucial tool in working with geometric sequences. It allows you to find any term in the sequence without having to calculate all previous terms, which can save a lot of time.

The standard form of the explicit formula for a geometric sequence is: \[ a_n = a_1 \cdot r^{(n-1)} \] where:

• \(a_n\) represents the \(n\)th term of the sequence,
• \(a_1\) is the first term,
• \(r\) is the common ratio,
• \(n\) is the position of the term within the sequence.

By plugging the given values for \(a_1\) and \(r\), and the desired term number, \(n\), into this formula, you can calculate exactly what any term in the sequence will be.

Common Ratio

The common ratio, denoted by \(r\), is a defining property of any geometric sequence. It is the factor that each term is multiplied by to get the next term in the sequence.

For example, if you know the first term \(a_1\) and the common ratio \(r\), you can determine subsequent terms quickly:

• The second term is found by multiplying the first term by the common ratio: \(a_2 = a_1 \cdot r\).
• The third term can be found by multiplying the second term by the ratio again: \(a_3 = a_2 \cdot r = a_1 \cdot r^2\).

Knowing the common ratio allows you to derive an entire geometric sequence from a single term. It is essential to understand this concept to work effectively with geometric sequences.

Nth Term of a Sequence

The \(n\)th term of a sequence refers to any term located at a specific position in that sequence. For geometric sequences, each term can be directly calculated using the explicit formula. This eliminates the need to sequentially calculate each preceding term.

To find the \(n\)th term, use: \[ a_n = a_1 \cdot r^{(n-1)} \] This formula shows how to determine the value of any term, \(a_n\).

In our example, given \(a_1 = -7\) and \(r = 0.1\), we calculated the first three terms:

• \(a_1 = -7\)
• \(a_2 = -7 \cdot 0.1 = -0.7\)
• \(a_3 = -7 \cdot 0.01 = -0.07\)

This method is especially helpful when needing to find large numbered terms without computing all previous ones.