Problem 77 - A Graphical Approach to Precalculus with Limits

Question

Determine the largest value of \(n\) that satisfies the inequality. $$\sum_{i=1}^{n}(2 k-1)<26$$

Step-by-Step Solution

Answer

The largest value of \( n \) is 5.

Step 1: Understand the sequence

The sequence given by \( \sum_{i=1}^{n}(2k-1) \) is equivalent to summing up the first \( n \) odd numbers. This can also be written as the sum of the first \( n \) odd integers: 1, 3, 5, 7,... etc.

Step 2: Use the formula for the sum of odd numbers

There is a formula for sum of the first \( n \) odd numbers: \( n^2 \). This means that \( \sum_{i=1}^{n}(2k-1) = n^2 \). We use this simplification to address the inequality \( n^2 < 26 \).

Step 3: Solve the inequality \( n^2 < 26 \)

To solve for \( n \), we need to find the largest integer. Solve \( n^2 < 26 \) by taking the square root of both sides: \( n < \sqrt{26} \). The square root of 26 is approximately 5.099, so \( n \) must be a whole number less than 5.099.

Step 4: Determine the largest integer \( n \) that satisfies the inequality

Since \( n \) must be an integer, the largest integer less than 5.099 is 5.

Step 5: Verify your solution

Calculate \( n^2 \) for \( n = 5 \), which gives \( 25 \), and check that \( 25 < 26 \). For \( n = 6 \), \( n^2 = 36 \) which does not satisfy the inequality since \( 36 > 26 \). Thus, the largest value of \( n \) is indeed 5.

Key Concepts

Sequences in Mathematics

A sequence in mathematics is a collection of numbers arranged in a specific, orderly manner. Each number in a sequence is called a term, and sequences are often categorized based on the pattern or rule that defines them. For example, a simple sequence could be the list of even numbers: 2, 4, 6, 8, and so forth. In the context of the exercise, the sequence of interest is the sequence of odd numbers, like 1, 3, 5, 7, etc.
This sequence of odd numbers is defined by a simple rule: each term differs from the previous one by a constant amount, specifically by adding 2. Understanding how sequences work is crucial when dealing with problems involving sums of series, where you're often tasked with adding up terms according to a specific rule.

Sum of Odd Numbers

The sum of the first few odd numbers has an interesting characteristic: it can be easily expressed using the formula \( n^2 \), where \( n \) is the number of terms to be added. For instance, if you sum the first 4 odd numbers: 1, 3, 5, and 7, you'll find that their sum is 16, which equals \( 4^2 \). This pattern holds true no matter how many terms you choose to add.
Utilizing the formula \( n^2 \) simplifies calculations significantly, especially when the total sum needs to be expressed as an algebraic inequality. For example, in the problem given, the task was to find the maximum \( n \) such that \( n^2 < 26 \). By setting up the inequality using this formula, solving it becomes straightforward and efficient.

Understanding Square Roots

Square roots play a key role in solving equations and inequalities, like \( n^2 < 26 \). The square root of a number is a value that, when multiplied by itself, gives the original number. For instance, the square root of 9 is 3, because \( 3 \times 3 = 9 \).
In the exercise, finding the square root of 26 helps to determine the largest integer \( n \) that meets the criteria \( n^2 < 26 \). When you compute \( \sqrt{26} \), the result is approximately 5.099. Since \( n \) must be a whole number, you would take the largest whole number less than 5.099, which is 5. Recognizing when and how to use square roots can aid in a smoother problem-solving process and help verify solutions.