Problem 13 - Algebra 2

Question

Write the explicit formula for each sequence. Then generate the first five terms. $$ a_{1}=5, r=-3 $$

Step-by-Step Solution

Answer

The explicit formula for the sequence is \(a_n = 5 + (n - 1) * -3\). The first five terms of the sequence are 5, 2, -1, -4, -7.

Step 1: Find the explicit formula

Given \(a_1=5\), and \(r=-3\), so the explicit formula for the sequence is \(a_n = a_1 + (n-1) * r = 5 + (n - 1) * (-3)\)

Step 2: Find the first term

To find the first term simply substitute \(n=1\) into the formula, then \(a_1 = 5 + (1 - 1) * -3 = 5\)

Step 3: Find the second term

Similarly, to find the second term substitute \(n=2\) into the formula, then \(a_2 = 5 + (2 - 1) * -3 = 2\)

Step 4: Find the third term

For the third term, with \(n=3\), we get \(a_3 = 5 + (3 - 1) * -3 = -1\)

Step 5: Find the fourth term

Substitute \(n=4\) into the formula to get the fourth term, which results in \(a_4 = 5 + (4 - 1) * -3 = -4\)

Step 6: Find the fifth term

Finally, calculate the fifth term by substituting \(n=5\) into the formula, so \(a_5 = 5 + (5 - 1) * -3 = -7\).

Key Concepts

Explicit Formula

An explicit formula is a powerful tool in mathematics, especially when dealing with sequences. It allows you to find any term in a sequence without having to list all the preceding terms. In the exercise, we are asked to find the explicit formula for an arithmetic sequence where the first term is given as 5 and the common difference \( r \) is -3.

The explicit formula for an arithmetic sequence can be written as:

• \( a_n = a_1 + (n-1) \cdot r \)

Here, \( a_n \) is the \( n^{th} \) term, \( a_1 \) is the first term, \( n \) is the term number, and \( r \) is the common difference between terms.

Calculating this using our given values, the explicit formula becomes:

• \( a_n = 5 + (n-1) \cdot (-3) \)

This formula provides a direct way to calculate the value of any term in the sequence just by plugging in the term number \( n \). This means no more crafting the previous terms if you're just interested in a specific one down the line.

Sequence Formula

The sequence formula is another name for the explicit formula described in the previous section, used to generate the terms of the sequence. This formula is especially handy when working with arithmetic sequences, which are characterized by a common difference between successive terms.

Let's see how the sequence formula is applied in practice by looking at the sequence with \( a_1=5 \) and \( r=-3 \):

• \( a_1 = 5 \)
• \( a_n = 5 + (n-1) \cdot (-3) \)

This sequence formula encapsulates the pattern of the sequence, allowing you to generate each term systematically. The formula's dependency on \( n \) demonstrates how flexible it is. Whether it's the second, tenth, or hundredth term, plugging \( n \) into this formula will yield your answer.

For students and problem solvers, understanding this as a formula provides a gateway to simplified arithmetic computations.

First Five Terms

Generating the first five terms of a sequence using an explicit formula is a straightforward process that reinforces the understanding of how sequences work. By substituting the values of \( n \) (from 1 to 5) into the explicit formula, we can find the first five terms of the sequence.

Here's how it is executed:

• First term: \( a_1 = 5 + (1-1) \cdot (-3) = 5 \)
• Second term: \( a_2 = 5 + (2-1) \cdot (-3) = 2 \)
• Third term: \( a_3 = 5 + (3-1) \cdot (-3) = -1 \)
• Fourth term: \( a_4 = 5 + (4-1) \cdot (-3) = -4 \)
• Fifth term: \( a_5 = 5 + (5-1) \cdot (-3) = -7 \)

These calculations illustrate the simplicity and efficiency of using the formula to pinpoint values quickly. It also demonstrates the nature of an arithmetic sequence where each term is adjusted by the common difference of -3, creating a linearly progressing set of terms.