Problem 59 - Algebra and Trigonometry

Question

Express each sum using summation notation. \(1+2+3+\cdots+20\)

Step-by-Step Solution

Answer

\(\sum_{n=1}^{20} n\)

Step 1 - Identify the Sequence

Observe the given sequence: 1, 2, 3, ..., 20. It is an arithmetic sequence where each term increases by 1.

Step 2 - Define the General Term

The general term of this arithmetic sequence can be expressed as: \(a_n = n\), where \(n\) represents the position of the term within the sequence.

Step 3 - Determine the Range of the Summation

The sequence starts at 1 and ends at 20. Therefore, the index \(n\) ranges from 1 to 20.

Step 4 - Construct the Summation Expression

Combine the general term and the range of the summation to express the sum in summation notation: \(\sum_{n=1}^{20} n\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which the difference between successive terms is constant. This difference is called the common difference. For example, in the sequence 1, 2, 3, ..., 20, each term increases by 1. So, the common difference here is 1.

Here are some key features of an arithmetic sequence:

• First term \(a_1\)
• Common difference \(d\)

Understanding these features helps us describe the terms of the sequence clearly.

General Term

The general term of an arithmetic sequence gives us a formula to find any term based on its position (or index) in the sequence. For any arithmetic sequence, the general term is given by: \[ a_n = a_1 + (n - 1) \cdot d \] Where:

• \(a_n\) is the \(n\)th term
• \(a_1\) is the first term
• \(d\) is the common difference
• \(n\) is the term's position in the sequence

For the sequence 1, 2, 3, ..., 20, the first term \(a_1 = 1\), and the common difference \(d = 1\). Plugging in the values, we get: \[ a_n = 1 + (n - 1) \cdot 1 = n \] Therefore, the general term for this sequence is \(a_n = n\).

Index Range

The index range specifies the starting and ending values for the indices of the terms you're summing. For example, in the sequence 1 to 20, the index starts at 1 and ends at 20.

In summation notation, if \(n\) is the index variable:

• The starting value of the index is the lower limit
• The ending value of the index is the upper limit

For our example sequence, the index \(n\) ranges from 1 to 20. Therefore, the index range is from 1 to 20.

Summation Expression

A summation expression compactly represents the sum of a sequence. It’s written using the Greek letter sigma \((\sum)\), which denotes summation. The general format is: \[ \sum_{i=m}^{n} f(i) \] Where:

• \( i \) is the index of summation
• \( m \) is the lower limit
• \( n \) is the upper limit
• \( f(i) \) is the function of \( i \) giving the terms of the sequence

For our sequence 1, 2, 3, ..., 20, using \( n \) as the index, the summation expression is: \[\sum_{n=1}^{20} n \]