Problem 33 - Beginning and Intermediate Algebra

Question

Find the indicated term for each arithmetic sequence. $$a_{1}=-7, d=-5 ; a_{21}$$

Step-by-Step Solution

Answer

The 21st term of the given arithmetic sequence is \(a_{21} = -107\).

Step 1: Understand the arithmetic sequence formula

In an arithmetic sequence, any term, denoted by \(a_n\), can be found using the formula: \[a_n = a_1 + (n-1)d\] where \(a_1\) is the first term, \(n\) is the number of terms, and \(d\) is the common difference between the terms.

Step 2: Substitute the given values

We are given \(a_1 = -7\), \(d = -5\), and we need to find \(a_{21}\). Substitute these values into the formula: \[a_{21} = -7 + (21 - 1)(-5)\]

Step 3: Simplify the expression

Now, we just need to simplify the expression to get the value for \(a_{21}\): \[a_{21} = -7 + (20)(-5)\] \[a_{21} = -7 - 100\] \[a_{21} = -107\] The 21st term of the given arithmetic sequence is -107.

Key Concepts

Common Difference

In an arithmetic sequence, the common difference is a crucial concept that defines the pattern of the sequence. This difference, denoted as \( d \), is consistent between each pair of consecutive terms in the sequence. To find the common difference, you simply subtract the previous term from the current term. For example, if the sequence starts as \( -7, -12, -17, \) then:

• The common difference \( d \) is \( -12 - (-7) = -5 \).

The negative sign indicates that the sequence is decreasing. Remember, what makes arithmetic sequences special is this constant increment or decrement, which can easily be identified by calculating and confirming the common difference between any two consecutive terms.

Sequence Formula

The sequence formula for an arithmetic progression is a very handy tool for locating any term within the sequence. It is expressed as:\[ a_n = a_1 + (n-1)d \]Where:

• \( a_n \) is the term you are trying to find,
• \( a_1 \) is the first term of the sequence,
• \( n \) is the position of the term in the sequence,
• \( d \) is the common difference.

This formula provides a straightforward path to finding the desired term without having to write out all the terms beforehand. It’s an efficient way to handle sequences and illustrates how arithmetic progression operates through a linear pattern.

Finding Terms

Finding terms in an arithmetic sequence is straightforward when using the sequence formula. For example, let's find the 21st term (\( a_{21} \)) for the sequence with a first term \( a_1 = -7 \) and a common difference \( d = -5 \). We substitute these values into the formula:\[ a_{21} = -7 + (21-1)(-5) \]After solving the equation, we simplify it step by step:

• Calculate \( 21-1 \) which equals 20.
• Then multiply 20 by the common difference \( -5 \) to get \( -100 \).
• Add to the first term \( -7 \), giving us \( -7 - 100 = -107 \).

Thus, the 21st term is \( -107 \). This systematic approach allows you to find any term in the sequence efficiently.

Arithmetic Progression

An arithmetic progression (AP) is a sequence of numbers where the difference between any two consecutive terms is constant. This progression features a fixed addition or subtraction pattern, easily described by its common difference \( d \). The nature of arithmetic progressions makes them predictable.
For instance, suppose we start with a sequence \( -7, -12, -17, ... \), continuing in this manner. The common difference here is \( -5 \), repeating with each step. As a result, the terms decrease neatly each time this difference is applied.
Understanding arithmetic progressions can significantly help in the fields of algebra and beyond, as they model real-world scenarios such as finance, science, and much more, where consistent changes take place over time.