Problem 62 - Algebra and Trigonometry

Question

\(61-68\) Write the sum using sigma notation. $$ 2+4+6+\dots+20 $$

Step-by-Step Solution

Answer

\( \sum_{i=1}^{10} (2i) \)

Step 1: Identify the Arithmetic Sequence

The sequence given is 2, 4, 6, ..., 20. This is an arithmetic sequence where each term increases by 2 (the common difference).

Step 2: Determine the First Term (a) and Common Difference (d)

The first term, \(a\), of the arithmetic sequence is 2. The common difference, \(d\), is 2, since each term increases by 2.

Step 3: Determine the Last Term (l)

The last term given in the sequence is 20.

Step 4: Calculate the Number of Terms (n)

Use the formula for the \(n\)-th term of an arithmetic sequence: \(a_n = a + (n - 1) \cdot d\). Substitute \(a_n = 20\), \(a = 2\), and \(d = 2\): \(20 = 2 + (n-1) \cdot 2\). Solve for \(n\):\[ 20 = 2 + 2(n-1) \]\[ 20 = 2 + 2n - 2 \]\[ 20 = 2n \]\[ n = 10 \].This means there are 10 terms in the sequence.

Step 5: Write the Sum in Sigma Notation

The sum of the sequence can be written in sigma notation as: \[ \sum_{i=1}^{10} (2i) \].Each term in the sequence is of the form \(2i\), beginning with \(i = 1\) and ending with \(i = 10\).

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a list of numbers where each number, after the first, is obtained by adding a constant to the previous term. This constant is known as the "common difference." It is a fundamental concept in mathematics, particularly when dealing with sequences and series.

• **Definition:** An arithmetic sequence is defined as a sequence of numbers in which the difference between consecutive terms is always the same.
• **Example:** Consider the sequence 2, 4, 6, 8, ..., 20. This is an arithmetic sequence where each term is consistently added with a difference of 2.

The role of an arithmetic sequence is crucial in understanding how a series behaves and is calculated. Such sequences often appear in real-world scenarios such as calculating regular savings, planning events, or distributing resources evenly.

Exploring the Common Difference

The common difference is a critical element in an arithmetic sequence. It defines the uniform pace at which the sequence progresses. Recognizing the common difference helps to quickly generate or verify terms within the sequence.

• **Identification:** To identify the common difference, simply subtract any term from the succeeding term in the sequence. For the sequence 2, 4, 6, the difference is 2, calculated as 4 - 2.
• **Constant Nature:** The common difference remains unchanged throughout the sequence, which means consistency is key to arithmetic sequences.

A strong grasp of the common difference allows for predicting future terms and understanding the sequence's structure. This makes problem-solving and calculations much more straightforward.

Using the nth Term Formula

The nth term formula is a powerful tool in dealing with arithmetic sequences. It provides a quick way to find any term in the sequence without having to list all previous terms. The formula is: \[ a_n = a + (n - 1) imes d \] This formula is essential when you want to determine a specific term's value at any position within the sequence.

• **Components:*** In the formula, \(a\) is the first term, \(d\) is the common difference, and \(n\) is the position of the term in the sequence.
• **Application:** To find the 10th term of the sequence starting with 2 and having a common difference of 2, substitute \(a = 2\), \(d = 2\), and \(n = 10\) into the formula: \( a_n = 2 + (10 - 1) imes 2 = 2 + 18 = 20 \).

Utilizing the nth term formula enables efficient exploration of arithmetic sequences, enhancing your problem-solving capabilities across various mathematical applications.