Problem 51 - Precalculus: Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

The sum is 8.

Step 1: Understand the Expression Inside the Sum

The expression \(1 + (-1)^i\) is inside the summation. This expression changes based on the value of \(i\). If \(i\) is even, \((-1)^i = 1\), making the expression \(1 + 1 = 2\). If \(i\) is odd, \((-1)^i = -1\), making the expression \(1 + (-1) = 0\).

Step 2: Determine Values for Each Term in the Sum

Evaluate the expression \(1 + (-1)^i\) as \(i\) goes from 1 to 8. For odd \(i\) (1, 3, 5, 7), the expression evaluates to 0. For even \(i\) (2, 4, 6, 8), it evaluates to 2.

Step 3: Calculate the Total Number of Terms that Contribute to the Sum

Only the terms where \(i\) is even contribute to the sum. There are four even numbers from 1 to 8, which are 2, 4, 6, and 8.

Step 4: Compute the Sum of the Contributing Terms

Each contributing term is 2, and there are four of them: \(2 + 2 + 2 + 2\). Thus, the sum is \(4 \times 2 = 8\).

Key Concepts

Series

In mathematics, a series is the sum of the terms of a sequence. When used with summation notation, it represents a compact way to describe the addition of several terms that often follow a specific pattern. Consider the expression \(\sum_{i=1}^{8}\), which indicates that we are summing values as \(i\) ranges from 1 to 8.

• A series can be finite, like in this exercise, or infinite, extending indefinitely.
• Summation notation helps manage sums where there are many terms, especially when they follow a predictable rule or formula.

In this case, each term is based on the expression \(1 + (-1)^i\), which changes value depending on whether \(i\) is even or odd. Such observations are crucial for solving the sum and understanding the behavior of the series.

Even and Odd Integers

Integers are classified into even and odd based on their divisibility by 2. An integer is even if it can be divided by 2 without a remainder, and odd if it cannot. This characteristic plays a role in determining the behavior of terms in a series.

• Even integers include ...,-4, -2, 0, 2, 4, and so on.
• Odd integers are found in between the even ones, such as ..., -3, -1, 1, 3, and so forth.

In the given exercise, the expression \((-1)^i\) produces different results depending on whether \(i\) is even or odd:- When \(i\) is even, \((-1)^i = 1\), so the summand becomes \(1 + 1 = 2\).- When \(i\) is odd, \((-1)^i = -1\), leading to the summand \(1 + (-1) = 0\).Understanding these differences can simplify calculating the sum, as only even-indexed terms contribute in this example.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators. In mathematical problems, they often describe a pattern or rule for calculating further terms in a sequence or series.

• They form the building blocks of more complex mathematical ideas.
• An algebraic expression can be simplified or evaluated based on given values or conditions.

For instance, the expression \(1 + (-1)^i\) changes its value depending on \(i\)'s parity (even or odd). Simplifying it correctly allows you to evaluate each term as \(i\) changes.
In exercises like this one, examining how an algebraic expression transforms with different inputs is a crucial skill. It offers a window into understanding how different terms in summation affect the overall total. Mastery over algebraic expressions can profoundly impact your ability to solve more complex mathematical challenges confidently.