Problem 45 - 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: Understanding the Series

We want to find the sum of the series \(\sum_{i=1}^{8}\left[1+(-1)^{i}\right]\). This is a series where each term is determined by substituting \(i\) from 1 to 8 into the expression \(1+(-1)^{i}\).

Step 2: Evaluating the Expression for Each Term

Calculate \(1 + (-1)^i\) for each integer \(i\) from 1 to 8:- When \(i = 1\), \((-1)^i = -1\) so term is \(1 - 1 = 0\).- When \(i = 2\), \((-1)^i = 1\) so term is \(1 + 1 = 2\).- Continue this pattern for all \(i\) from 3 to 8.

Step 3: Calculate Each Term

Following the pattern:- \(i = 1\): \(1 + (-1)^1 = 0\)- \(i = 2\): \(1 + (-1)^2 = 2\)- \(i = 3\): \(1 + (-1)^3 = 0\)- \(i = 4\): \(1 + (-1)^4 = 2\)- \(i = 5\): \(1 + (-1)^5 = 0\)- \(i = 6\): \(1 + (-1)^6 = 2\)- \(i = 7\): \(1 + (-1)^7 = 0\)- \(i = 8\): \(1 + (-1)^8 = 2\)

Step 4: Summing Up the Terms

Add all the values calculated for \(i = 1\) to \(i = 8\):\(0 + 2 + 0 + 2 + 0 + 2 + 0 + 2 = 8\).

Key Concepts

Alternating Series

An alternating series is a sequence of numbers where the terms change signs, meaning they switch between positive and negative. In our exercise, the expression \((-1)^i\) is responsible for this alternation. When \(i\) is odd, \((-1)^i\) results in \(-1\), making the entire term zero since \(1 - 1 = 0\). On the other hand, when \(i\) is even, \((-1)^i\) gives \(1\), making the entire term equal to two, because \(1 + 1 = 2\).

To identify an alternating series:

• Look for a component like \((-1)^n\) which creates an alternating pattern.
• Notice how this pattern affects each term in the series.
• The position of terms (odd or even) directly influences the signs and values of terms.

By understanding these properties, you can easily identify and work with alternating series in other exercises.

Arithmetic Sequence

An arithmetic sequence is a series of numbers where the difference between consecutive terms is always the same. However, in the given exercise, while the alternating series is not an arithmetic sequence, it can still connect to concepts of repetition and pattern.

The sequence in this problem, with terms 0, 2, 0, 2, and so on, forms a repeating pattern. This repetition is a key aspect of understanding many sequences and simplifies the process of summation, as patterns can greatly reduce the necessity for tedious calculations.

When numbers in a sequence repeat:

• Identify the repeating block; in this exercise, it is "0, 2".
• Count how many times the block repeats within the sequence.
• Perform the necessary arithmetic operations on one block, then multiply by the number of repetitions.

Recognizing these patterns enables quicker computation and offers insight into sequence structures beyond typical arithmetic methods.

Precalculus Concepts

Precalculus forms the foundation of more advanced math topics like calculus and trigonometry. Understanding series and sequences, as seen in this exercise, is a core part of precalculus. These concepts help lay down the groundwork for dealing with infinite series and complex functions later on.

Key precalculus elements at play here include:

• The sum of a sequence, which is typically expressed using sigma notation \(\sum\).
• Understanding how expressions like \(1 + (-1)^i\) affect series.
• Recognizing alternating sequences and their patterns which are vital for future calculus challenges.

Precalculus ensures you are comfortable with mathematical ideas and notations, paving the way for smooth progression into calculus. By focusing on these foundational concepts, you develop problem-solving skills that are useful in various mathematical contexts.