Problem 15 - Calculus for Biology and Medicine

Question

Write each sum in sigma notation. $$ 2+4+6+8+\cdots+2 n $$

Step-by-Step Solution

Answer

The sum in sigma notation is \( \sum_{k=1}^{n} 2k \).

Step 1: Identify the Pattern

Look at the given sequence: 2, 4, 6, 8, ..., 2n. Notice that each term increases by 2, which is a constant difference. This is an arithmetic sequence.

Step 2: Find the General Term

Since this is an arithmetic sequence, the general term of the sequence can be expressed as \(a_n = a + (n-1)d\), where \(a\) is the first term and \(d\) is the common difference. Here, \(a = 2\) and \(d = 2\), so the general term \(a_n = 2 + (n-1) \times 2 = 2n\).

Step 3: Use Sigma Notation

Now that we have the general term \(a_n = 2n\), we can write the sum using sigma notation. The sequence sum is from the first term (\(n = 1\)) to the nth term. Therefore, the sigma notation is: \[ \sum_{k=1}^{n} 2k \]

Step 4: Confirm the Sigma Notation

Verify that the sigma notation represents the sequence correctly. For each \(k\) starting from 1 up to \(n\), \(2k\) generates the sequence 2, 4, 6, ..., 2n. This confirms our sigma notation is correct.

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a series of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. This constant is known as the 'common difference.' For example, in the sequence 2, 4, 6, 8,..., the common difference is 2. Arithmetic sequences are straightforward because they have a consistent pattern that can be easily spotted. In context, recognizing this pattern helps us in finding other elements of the sequence, including the general term or the sum of the sequence's elements.

The General Term of an Arithmetic Sequence

The general term, often denoted as \(a_n\), allows us to find any term in an arithmetic sequence without listing all terms sequentially. For an arithmetic sequence with an initial term \(a\) and a common difference \(d\), the general term formula is given by: \[ a_n = a + (n-1) imes d \] This formula works by allowing us to count from the start, adding the common difference the appropriate number of times. In the sequence 2, 4, 6, ..., by using \(a = 2\) (the first term) and \(d = 2\), we form the general term: \[ a_n = 2 + (n-1) imes 2 = 2n \] This expression indicates multiplies the position of the term \(n\) by 2 to find its value within the sequence.

How to Find the Sequence Sum Using Sigma Notation

To find the sum of a sequence, especially an arithmetic one, we employ the power of sigma notation. Sigma notation is a compact and efficient way to represent the summation of a series of terms. For an arithmetic sequence, it's particularly useful because it simplifies the process of adding up many terms. In our example with the sequence 2, 4, 6, ..., 2n, the terms can be represented individually as 2, 4, 6, and so on up to 2n. Using the general term \(2n\), the sequence sum (or total) can be expressed in sigma notation as: \[ \sum_{k=1}^{n} 2k \] This equation indicates that as \(k\) varies from 1 to \(n\), each term of the form \(2k\) is summed. Sigma notation thus elegantly represents the process of adding each term from the first to the nth.

The Power of Summation in Sequences

Summation is the concept of adding a series of numbers together. In sequences, particularly arithmetic ones, summation allows us to compute the total value of a sequence quickly and systematically. Using the sigma notation, summarizing an entire series becomes a concise task because it provides a formulaic approach rather than creating a long arithmetic expression. By using \( \sum \), we indicate that a sum is taking place, where each term of the sequence follows the governing formula or rule (for an arithmetic sequence, it's typically something simple involving the term's position). This approach allows not only for quick calculations but also aids in understanding and verifying the sequence and its properties, ensuring accuracy and clarity in mathematical explorations.