Problem 43 - Algebra and Trigonometry

Question

Find the sum. $$\sum_{i=1}^{8}\left[1+(-1)^{i}\right]$$

Step-by-Step Solution

Answer

The sum is 8.

Step 1: Define the Series Trm

First, identify the expression inside the summation: \(1+(-1)^{i}\). This formula depends on the value of \(i\): when \(i\) is an odd number, \((-1)^{i} = -1\); when \(i\) is an even number, \((-1)^{i} = 1\). Use this information to calculate the series term.

Step 2: Calculate the Term for Odd \(i\)

For odd values of \(i\), e.g., \(i=1, 3, 5, 7\), we have: \[1+(-1)^i = 1 - 1 = 0\]. This shows that the term equals zero for these cases.

Step 3: Calculate the Term for Even \(i\)

For even values of \(i\), e.g., \(i=2, 4, 6, 8\), we have: \[1+(-1)^i = 1 + 1 = 2\]. This shows that the term equals two for these cases.

Step 4: Calculate the Sum of the Series

The series is composed of the terms from when \(i\) is odd and even. For odd \(i\) (1, 3, 5, 7), the term contributes nothing (0). For even \(i\) (2, 4, 6, 8), each term contributes a value of 2. Therefore, the total number of even terms is 4: \((4 \times 2 = 8)\).

Step 5: Verify the Result

Verify by explicitly adding each term: for \(i = 1\), the term is \(0\); for \(i = 2\), the term is \(2\); for \(i = 3\), the term is \(0\); for \(i = 4\), the term is \(2\); for \(i = 5\), the term is \(0\); for \(i = 6\), the term is \(2\); for \(i = 7\), the term is \(0\); and for \(i = 8\), the term is \(2\). Adding these, \(0 + 2 + 0 + 2 + 0 + 2 + 0 + 2 = 8\).

Key Concepts

Summation Notation

Understanding summation notation is crucial when dealing with series. It is represented by the Greek letter Sigma (\(\Sigma\)) and is used to denote the sum of a sequence of terms. The notation \(\sum_{i=1}^{8}(...)\) tells us to sum the specified expression as the variable \(i\) ranges from the lower limit of 1 to the upper limit of 8.
Here's what each part means:

• \(i\): The index of summation, which takes on successive integer values from the lower limit to the upper limit.
• 1 to 8: These numbers indicate the range over which you sum the expression.
• Expression inside brackets \([1+(-1)^{i}]\): Defines the terms that are being summed.

To simplify this concept, imagine it as a set of instructions. These instructions guide you to identify each term in the series, calculate its value, and then sum all these values together.

Even and Odd Numbers

A series often relies on whether numbers are even or odd, influencing their contribution to the sum. An odd number is an integer not divisible by 2, leaving a remainder of 1 when divided by 2. Examples include 1, 3, 5, and so on. Conversely, an even number divides evenly by 2, leaving no remainder, such as 2, 4, 6, etc.
In the context of our series, understanding whether \(i\) is even or odd is crucial because it determines the term's value:

• When \(i\) is odd (e.g., 1, 3, 5, 7), the expression \((-1)^{i}\) becomes -1, leading to a term value of 0 \([1 + (-1) = 0]\).
• When \(i\) is even (e.g., 2, 4, 6, 8), the expression \((-1)^{i}\) becomes 1, resulting in a term value of 2 \([1 + 1 = 2]\).

This interplay between even and odd numbers allows us to systematically approach calculating the series.

Series Calculation

To calculate a series, follow a structured approach to find each term and their total sum. Start by identifying whether each term is for an even or odd \(i\). In our example:

• For odd \(i\) values like 1, 3, 5, and 7, the term automatically turns into 0.
• For even \(i\) values like 2, 4, 6, and 8, each contributes 2 to the sum.

Next, sum all the terms you've calculated. Here, since there are four even indices (2, 4, 6, 8) each contributing 2, the sum is calculated as:2 (for \(i=2\)) + 2 (for \(i=4\)) + 2 (for \(i=6\)) + 2 (for \(i=8\)) = 8.
This straightforward enumeration and addition process helps in systematically solving series problems with ease.

Summation Formula

A summation formula expresses the total of a sequence, particularly useful in simplifying calculations of long series. Typically, these formulas can help you calculate the sum of series without addressing each term individually.In our exercise's context, given that the sum involves differentiating terms based on whether the index is even or odd, you don't need a typical arithmetic or geometric series formula. The logic of separating terms into zero or some consistent value (in this case, 2) per index is sufficient.
Still, understanding basic summation formulas can be beneficial:

• Arithmetic Series: For an arithmetic series, the sum is given by \(S = \frac{n}{2} (a + l)\), where \(n\) is the number of terms, \(a\) is the first term, and \(l\) is the last term.
• Geometric Series: For a geometric series, the sum is given by \(S = a\frac{1-r^n}{1-r}\), provided \(|r| < 1\).

By learning to apply these formulas in relevant contexts, you can enhance your problem-solving skills for more complex series calculations.