Problem 24 - Beginning and Intermediate Algebra

Question

Given the arithmetic sequence \(-9,-21,-33,-45,-57, \dots\) a) Find \(a_{1}\) and \(d\) b) Find a formula for the general term of the sequence, \(a_{n}\) c) Find \(a_{15}\)

Step-by-Step Solution

Answer

a) \(a_1 = -9\), \(d = -12\) b) \(a_n = -9 + (n-1)(-12)\) c) \(a_{15} = -177\)

Step 1: Find \(a_1\) and \(d\) from the sequence

To find the first term \(a_1\), we can simply look at the first number in the given sequence. Thus, \(a_1 = -9\). Next, let's find the common difference \(d\). We need to find the difference between consecutive terms in the sequence; we can do this by subtracting the first term from the second term: \[d = (-21) - (-9) = -21 + 9 = -12\] In the given sequence, \(a_{1} = -9\) and the common difference \(d = -12\).

Step 2: Find the general term formula

Now that we have the first term and the common difference, we can write the general term formula, \(a_n\), as follows: \[a_n = a_{1} + (n-1)d\] Substituting the values of \(a_1\) and \(d\): \[a_n = (-9) + (n-1)(-12)\] This is the formula for the general term of the given sequence.

Step 3: Find the \(15^{th}\) term \(a_{15}\)

To find the \(15^{th}\) term of the sequence, we need to substitute \(n = 15\) in the general term formula we derived in step 2: \[a_{15} = (-9) + (15-1)(-12)\] Now, let's calculate the value of \(a_{15}\): \[a_{15} = (-9) + (14)(-12)\] \[a_{15} = -9 - 168\] \[a_{15} = -177\] Hence, \(a_{15} = -177\).

Key Concepts

Common Difference

When working with arithmetic sequences, understanding the common difference is essential. The common difference, denoted as \(d\), is the consistent interval or difference between consecutive terms of an arithmetic sequence.
This means if you pick any two successive elements in the sequence and subtract the earlier term from the later one, you should always get the same value.
For example, in the arithmetic sequence \(-9, -21, -33, -45, -57, \ldots\), the common difference \(d\) can be found by subtracting the first term from the second one:

• Second term: \(-21\)
• First term: \(-9\)
• Common difference: \(-21 - (-9) = -12\)

Finding the common difference helps to identify the pattern of the sequence and is a necessary step in deriving further sequence terms.

General Term

The general term of an arithmetic sequence provides a formula to find any term in the sequence without having to write out all previous terms. It is usually expressed as \(a_n\), where \(n\) is the position of the term within the sequence.
The formula for the general term is:
\[a_n = a_1 + (n-1)d\]
Where:

• \(a_1\) is the first term in the sequence.
• \(d\) is the common difference.
• \(n\) is the term position you want to find.

For the sequence given \((-9, -21, -33, \ldots)\), substituting \(a_1 = -9\) and \(d = -12\) into the formula gives:
\[a_n = -9 + (n-1)(-12)\]
This allows you to determine any term directly by substituting \(n\) with the desired term's position.

Sequence Formula

The sequence formula in arithmetic sequences enables us to find any particular term systematically without listing all terms up to that position.
It's derived using the general term formula, which is:
\[a_n = a_1 + (n-1)d\]
In the example sequence \(-9, -21, -33, \ldots\), we calculated:

• \(a_1 = -9\), the first term
• \(d = -12\), the common difference

Inserting the known values gives the sequence formula:
\[a_n = -9 + (n-1)(-12)\]
This formula lets anyone find, say the 15th term, by simply substituting \(n=15\):
1. \(a_{15} = -9 + (15-1)(-12)\)2. Simplifying further: \[a_{15} = -9 - 168 = -177\]
Using this process, the 15th term, \(-177\), was calculated efficiently without manually writing out all preceding terms.