Problem 47 - College Algebra

Question

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1+2+3+\dots+30$$

Step-by-Step Solution

Answer

The summation notation for the given sequence is \(\sum_{i=1}^{30} i\).

Step 1: Identify the pattern and the index

Given the sequence, 1, 2, 3, ..., 30, it can be seen that the pattern increases by 1 each time. The index of summation \(i\) in this case, which represents each term in the sequence, will be equal to the term's position within the sequence itself.

Step 2: Set the lower and upper limit

As per the problem, use \(I\) as the lower limit of summation and \(i\) for the index of summation. Since the sequence starts at 1 and ends at 30, the lower limit \(I\) is equal to 1 and the upper limit is 30.

Step 3: Write the sum in summation notation

The summation notation will be \(\sum_{i=I}^{30} i\).

Key Concepts

Arithmetic Sequence

An arithmetic sequence, also known as an arithmetic progression, is a series of numbers in which each term after the first is formed by adding a constant difference to the previous term. In simpler terms, it’s a list of numbers where the difference between consecutive numbers remains the same.
For example, in the sequence given in the exercise: \(1, 2, 3, \ldots, 30 \), each number increases by 1 from the previous number. This regular increase makes it an arithmetic sequence with a common difference of 1.

• First Term, \(a_1 = 1\)
• Common Difference, \(d = 1\)
• Maintains consistent changes throughout the sequence

Arithmetic sequences are straightforward but form the basis for understanding more complex patterns in mathematics. Recognizing these patterns can greatly simplify finding terms or sums within the series.

Index of Summation

The index of summation is a variable often used as a placeholder in summation notation, allowing us to compactly express long sums. This index typically appears in the form of a letter, most commonly "i","j", or "k", and it represents each term within the summation. A good analogy is thinking of the index like a loop counter in programming. It keeps track of which term in the sequence you're currently summing.
In our example, the index of summation is "i", which denotes each term in the arithmetic sequence from 1 to 30 within the summation notation \( \sum_{i=I}^{30} i \).

• Allows compact writing of long sums
• Identifies the current position in the sequence
• Helps iterate through sequence to include each applicable term

Thus, identifying and understanding the index of summation is fundamental for using summation notation effectively.

Limits of Summation

The limits of summation are the starting and ending values for the index in summation notation, which tell us the range over which the sum is calculated. These limits are crucial because they specify the exact part of the sequence being summed, much like specifying which sections of a book you intend to read.
In the exercise, the problem specifies using "I" as the lower limit. However, to stay consistent and correct, it can be interpreted in lowercase as the clarification of the sequence starting point. For this sequence:

• Lower Limit, \(I = 1\): The sequence starts from the first term, number 1.
• Upper Limit, \(30\): Indicates the sequence ends at the number 30.

The limits direct the scope of our summation. Thus, when you see \( \sum_{i=1}^{30} i \), it denotes that each term starting from 1 up to 30 is included in the sum.