Problem 53 - College Algebra

Question

Express each sum using summation notation. Use I as the lower limit of summation and i for the index of summation. $$1+3+5+\dots+(2 n-1)$$

Step-by-Step Solution

Answer

\(\sum_{i=1}^{n}(2i-1)\

Step 1: Identify pattern

Identify the pattern that the series is following. Here, we have an arithmetic series of odd numbers, where each term can be represented as (2i-1), where i is the term number.

Step 2: Define Limits for Summation

Since we start from 1 (the first term), the lower limit for summing these numbers will be i=1. The upper limit will be n, as the series goes up to the nth term.

Step 3: Write the Summation

Now, write the summation notation using the pattern identified in step 1 and the limits defined in step 2. The result will be \(\sum_{i=1}^{n}(2i-1)\)

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term after the first is obtained by adding a constant difference to the preceding term. In simple terms, it involves numbers that increase (or decrease) by the same amount. The sequence given in the exercise, like 1, 3, 5, ..., follows a simple rule: each subsequent number is obtained by adding 2 to the previous one.

• First term: 1
• Second term: 1 + 2 = 3
• Third term: 3 + 2 = 5

This constant addition is known as the common difference. In this particular sequence, the common difference is 2. When expressed using summation notation, an arithmetic series allows us to compactly represent the sum of the series using special mathematical symbols.

Index of Summation

The index of summation, often represented by a letter such as 'i', is a very important part of summation notation. It plays the role of a counter, helping to specify which terms are being added up in a series. In the exercise solution, 'i' is used:

• It starts from the lower limit, meaning it begins counting from the first term in the series.
• It increases by 1 each time until it reaches the upper limit, covering all terms in the sequence.

The index of summation is particularly useful when expressing large series, as it succinctly indicates both the starting point and how each term is derived by the index itself.

Upper Limit

The upper limit in summation notation defines where the series stops. In other words, it's the ending point for the summation index. In a sense, it tells us how many terms to include in the sum. Let's revisit the exercise:

• Here, the upper limit is 'n', indicating that the series continues up to the nth term.
• This upper limit ensures that the summation covers all intended terms, right up to the final specified number in the series.

By setting an upper limit, we have control over exactly how many terms are summed, offering clarity and precision to the representation.

Lower Limit

The lower limit in summation notation marks the beginning point from which summation starts. It sets the baseline number or index from which we commence counting and summing the series terms.

• In the given series, the lower limit is set at 'i=1'.
• This signifies that the summation starts with the first term.

Setting a lower limit is crucial because it determines the starting point for the index of summation, ensuring that the notation accurately represents the entire series from the first term onward. It makes the notation clean and avoids any ambiguity about where the summation should commence.