Problem 41 - 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 first 60 positive even integers.

Step-by-Step Solution

Answer

The 20th term in the sequence is 40 and the sum of the first 60 positive even integers is 3660.

Step 1: Formula for the nth term of an arithmetic sequence

The formula for the nth term (\(a_{n}\)) of an arithmetic sequence is given by \(a_{n} = a_{1} + (n-1)d\), where \(a_{1}\) is the first term and d is the common difference. For the arithmetic sequence of positive even numbers (\(a_{1}=2, d=2\)), the formula becomes \(a_{n} = 2 + (n-1)2 = 2n\).

Step 2: Find the 20th term

Insert n=20 into the formula from step 1 to find the 20th term. Thus, \(a_{20} = 2*20 = 40\).

Step 3: Sum of the first 60 positive even integers

The sum of the first 60 terms of an arithmetic sequence is given by \(S_{n} = n/2 * (a_{1} + a_{n})\), where \(S_{n}\) is the sum of the first n terms, \(a_{1}\) is the first term, and \(a_{n}\) is the nth term. Substituting \(n=60, a_{1}=2, a_{60}=2*60\) into the formula, we get \(S_{60} = 60/2 * (2 + 120) = 30 * 122 = 3660\).

Key Concepts

nth term formula

Understanding the concept of the nth term is key to unraveling arithmetic sequences. The nth term formula provides a straightforward way to determine any term in an arithmetic sequence without continually adding the common difference. It is expressed as \(a_{n} = a_{1} + (n-1) \cdot d\), where \(a_{n}\) is the term you want to find, \(a_{1}\) is the first term in the sequence, and \(d\) is the common difference.
In simpler terms, after you know the first term and the gap between each term, you can calculate any specific term by plugging in the position of that term (n) into the formula. For example, the sequence of positive even integers starts with 2, and the difference between each term is 2. Thus, the nth term formula becomes \(a_{n} = 2n\). This simplicity makes it easy to find, say, the 20th term by just inputting \(n = 20\) to get \(a_{20} = 40\).

sum of arithmetic sequence

Calculating the sum of terms in an arithmetic sequence involves a handy formula: \(S_{n} = \frac{n}{2} \cdot (a_{1} + a_{n})\). This formula allows you to quickly determine the total sum of a contiguous set of terms. It is designed to leverage the arithmetic properties of the sequence, using the first and last terms (\(a_{1}\) and \(a_{n}\)) as well as the number of terms (n).

• \(S_{n}\) is the sum of the first n terms.
• \(a_{1}\) is the first term.
• \(a_{n}\) is the nth term.

In our example, you can find the sum of the first 60 positive even integers by identifying \(a_{60}\) as 120 (since \(a_{60} = 2 \cdot 60\)). Substituting \(n = 60, a_{1} = 2\), and \(a_{60} = 120\) into the formula gives us \(S_{60} = 30 \cdot 122 = 3660\). Thus, the swift calculation delivers the sum 3660 without needing to add each term individually.

common difference

The common difference is a distinction of arithmetic sequences. It represents the consistent change between successive terms. The common difference (d) thus defines the sequence's linear progression. Concisely, if you have an arithmetic sequence, each term after the first is simply the previous term plus the common difference.
For positive even integers, the common difference is 2, which means each consecutive even number is 2 more than the previous one (for instance, 2, 4, 6, 8, ...). Understanding this core concept allows you to comprehend the linear pattern and predict subsequent sequence numbers. Detecting the common difference helps pivot the use of formulas for computing both particular terms and sequence sums.

positive even integers

Positive even integers form an easily recognizable sequence where each term is two more than the previous. Generally beginning at 2, the sequence extends infinitely: 2, 4, 6, 8, and so on. These numbers are critically structured as arithmetic sequences, with a first term \(a_{1}\) of 2 and a common difference (d) of 2.
The attribute that sets positive even integers apart is their form: they can be expressed as \(2n\) for integer values of \(n\). This expression not only simplifies locating terms in a sequence but also eases understanding sums and interdependencies within the series. These even numbers provide a clear example of arithmetic sequences, reinforcing the consistent step of addition defined by the common difference.