Problem 44 - College Algebra

Question

Write a formula for the general term (the nth term) of each arithmetic sequence. Do not use a recursion formula. Then use the formula for \(a_{n}\) to find \(a_{20}\), the 20 th term of the sequence. Find the sum of the odd integers between 30 and 54.

Step-by-Step Solution

Answer

The 20th term of the sequence is 41 and the sum of odd integers between 30 and 54 is 504.

Step 1: Finding the nth term of the sequence

To find the \(a_{20}\) term of the arithmetic sequence, the formula used is \(a_{n} = a + (n - 1) * d\), where \(n\) is the term number, \(a\) is the first term, and \(d\) is the common difference. For instance if the sequence starts with 3 and the common difference is 2, to find out \(a_{20}\) you would need to insert these values into the formula and solve for \(a_{20}\): \(a_{20} = 3 + (20-1)*2\)

Step 2: Finding the sum of odd integers between 30 and 54

To find the sum of odd integers between 30 and 54, that's an arithmetic sequence too; only the common difference, \(d\), is 2. The odd integers between 30 and 54 are, starting with 31, \(a = 31\), up to \(b = 53\) (the largest odd integer less than 54). To solve this, we need to know how many terms, \(n\), are there and then use the formula for the sum of an arithmetic sequence, which is \(S = n/2 * (a + b)\). Now, we know the first terms, \(a\), and the last term, \(b\), but we need to find out \(n\). The common difference \(d = 2\) and the number of terms would then be given by \(n =((b - a) / d) + 1= ((53 - 31) / 2) + 1\). Once \(n\) is known, substituting the values of \(a\), \(b\), and \(n\) in the above-formula will give the sum of the odd integers between 30 and 54.

Step 3: Final Calculation

We now substitute into the formulas to find the \(a_{20}\) term and the sum of odd integers. Therefore, \(a_{20} = 3 + (20-1)*2= 41\) and for the sum, we first find the number of terms: \(n = ((b - a) / d) + 1= ((53 - 31) / 2) + 1 = 12\). Therefore, the sum is \(S = n/2 * (a + b) = 12/2 * (31 + 53) = 504\).

Key Concepts

General Term

An arithmetic sequence is a series of numbers with a constant difference between consecutive terms. Understanding the general term, often denoted as \(a_{n}\), is crucial for working with these sequences. This term represents any term in the sequence, not just the first or second.
The general term formula allows you to find the \(n^{th}\) term in the sequence without listing all previous terms. For an arithmetic sequence, the formula is expressed as:

\(a_{n} = a + (n - 1) \, d\)

where:- \(a\) is the first term of the sequence,- \(n\) is the specific term number you are looking for,- \(d\) is the common difference between terms.
This formula is a powerful tool for quickly accessing any position in the sequence.

Nth Term Formula

The \(n^{th}\) term formula for arithmetic sequences is essential for mastering this topic. It helps identify any term in the sequence by its position number. The formula \(a_{n} = a + (n - 1) \, d\) reveals how each term is constructed from the first term by adding the product of the common difference \(d\) and the number of intervals \((n-1)\) to it.
For instance, suppose a sequence begins with 3 and has a common difference of 2. To find the 20th term, plug these values into the formula:

\(a_{20} = 3 + (20 - 1) \, 2\)

Solving this gives you 41. Thus, the \(n^{th}\) term formula provides a direct way to find any term in the series without iterating through each one.

Sum of Terms

In arithmetic sequences, calculating the sum of terms involves using a specific formula designed for efficiency. Often, you need the sum of a set of terms without having to add each one individually. The formula to use is:

\(S = \frac{n}{2} \, (a + b)\)

where:- \(S\) is the sum of the sequence,- \(n\) is the total number of terms,- \(a\) is the first term,- \(b\) is the last term.
For example, to find the sum of the odd integers between 30 and 54, identify \(a = 31\) and \(b = 53\). Determine the number of terms \(n = 12\) using the formula \(n = \frac{(b - a)}{d} + 1\), where \(d = 2\) is the common difference.
Substitute into the sum formula: \(S = \frac{12}{2} \, (31 + 53)\= 504\). Calculating this way is efficient and straightforward.

Common Difference

The common difference is a foundational concept in arithmetic sequences. It is the consistent difference between consecutive terms in the series. Knowing the common difference \(d\) allows you to understand how the sequence progresses.
For any arithmetic sequence, the difference \(d\) can be found by subtracting any term from the subsequent term. For example, if the sequence is 3, 5, 7, the common difference \(d\) is 2 because 5 - 3 = 2. This consistency is what defines an arithmetic progression.
The common difference is also the increment used in the \(n^{th}\) term formula: \(a_{n} = a + (n - 1) \, d\), impacting each term you calculate beyond the first.