Problem 60 - Algebra and Trigonometry

Question

Write the sum using sigma notation. \(2+4+6+\cdots+20\)

Step-by-Step Solution

Answer

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

Step 1: Identify the pattern

Look at the series given: \(2+4+6+\cdots+20\). The series is an arithmetic sequence where each term increases by 2. We recognize that this sequence starts at 2 and ends at 20.

Step 2: Determine the general term of the sequence

The first term \(a_1\) is 2, and all subsequent terms increase by a common difference \(d\) of 2. The nth term of an arithmetic sequence is given by the formula \(a_n = a_1 + (n-1)\cdot d\). In this case, \(a_n = 2 + (n-1)\cdot 2 = 2n\).

Step 3: Find the number of terms

To determine the number of terms \(n\), we set the general term equal to the last term in the series, which is 20. \(2n = 20\). Solving for \(n\), we get \(n = 10\).

Step 4: Write the series in sigma notation

Since the sequence is \(2, 4, 6, \dots, 20\) with 10 terms and the general term \(2n\), we can write the series in sigma notation as \(\sum_{n=1}^{10} 2n\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers in which each number after the first is obtained by adding a constant difference to the previous number. This constant is known as the "common difference." It is important to recognize this pattern, as it helps in identifying the series.
For example, in the sequence given in the exercise, each term increases by 2: 2, 4, 6, and so on. This common difference of 2 allows us to understand how the sequence grows.

• First Term ( ): This is the initial term of the arithmetic sequence. In our exercise, it is 2.
• Common Difference ( d >): This is the fixed amount added to each term to get to the next term. Here, it is 2.

By identifying these key components, you can easily begin working through problems involving arithmetic sequences.

General Term

The general term of an arithmetic sequence is a formula that lets us find any term in the sequence without listing all preceding terms. This formula is essential, as it allows you to calculate any specific term efficiently.
The formula for the general term of an arithmetic sequence is given by:
\[ a_n = a_1 + (n - 1) imes d \]
Let's break down what each element means:

• : The th term of the sequence.
• : The first term in the sequence.
• ( n - 1 ): Represents the number of intervals between the first term and the nth term.
• d: The common difference between consecutive terms.

In the problem, substituting the values into the formula, you get \( a_n = 2 + (n - 1) imes 2 = 2n \), showing how each term relates to the term number .

Number of Terms

To determine the total number of terms in an arithmetic sequence, you can use the general term formula by equating it to the last term of the sequence. This step confirms how many terms are present until the sequence reaches its conclusion.
For example, using the last term given in the problem, which is 20, we solve the equation \(2n = 20\).

• First, divide by 2 to isolate : \[ n = \frac{20}{2} = 10 \]

This calculation tells us that there are 10 terms in the given sequence. Determining the number of terms is vital for properly setting up notation for the series, ensuring accurate calculations.

Series

A series is the sum of the terms of a sequence. In this context, arithmetic series result from summing the terms of an arithmetic sequence. Sigma notation is a concise way to represent this sum.
For the series given, we express it in sigma notation as:
\[ \sum_{n=1}^{10} 2n \]
In this expression, the sigma symbol \( \sum \) denotes summation. Here's how the components of the notation break down:

• The is set at 1, denoting the start of the summation with the first term.
• The is 10, corresponding to the number of terms.
• The is <2n>, which gives the individual terms from the sequence.

Using sigma notation simplifies representation of the sequence's total sum, streamlining calculations and providing a powerful tool for dealing with larger sequences.