Problem 14 - Calculus Single Variable

Question

In Exercises \(11-16,\) use summation notation to express the sum. $$ 9+16+25+36 $$

Step-by-Step Solution

Answer

The sum is written as \( \sum_{n=3}^{6} n^2 \).

Step 1: Identify the Sequence and Pattern

First, observe the sequence of numbers given: 9, 16, 25, 36. Notice that these are perfect squares: \(9 = 3^2\), \(16 = 4^2\), \(25 = 5^2\), \(36 = 6^2\). This indicates that the sequence can be expressed in terms of squared integers starting from 3 to 6.

Step 2: Define the Index Range

Identify the index range that covers the sequence. The sequence starts with\(3^2\) and ends with \(6^2\). Therefore, the index \(n\) should start from 3 and go up to 6.

Step 3: Set up the Summation Notation

Now, express the sum using summation notation. The general term is \(n^2\) where \(n\) ranges from 3 to 6. Thus, the sum can be written as: \[ \sum_{n=3}^{6} n^2 \]

Key Concepts

Sequence Identification

In mathematics, identifying the sequence within a series of numbers is crucial for understanding the pattern and structure behind it. In this exercise, the given numbers are 9, 16, 25, and 36. By examining each number closely, we recognize them as perfect squares. Here's how this works:

• 9 is expressed as \(3^2\); meaning 3 multiplied by itself.
• 16 as \(4^2\); 4 times 4.
• 25 as \(5^2\); 5 times 5.
• And finally, 36 as \(6^2\); where 6 multiplies itself.

Recognizing this pattern is vital. Once you see these numbers as squares, it's clear that they form a series of consecutive perfect squares: starting from 3 to 6. This realization simplifies the problem greatly because we can now use a formulaic expression to represent the sum of these squares. By clearly seeing the sequence, we prepare ourselves to neatly express it using summation notation.

Index Range

The index range plays a key role in expressing series using summation notation. In our example, after identifying that 9 to 36 are perfect squares, we need to determine the range that these squares cover. The first perfect square is 9, which is \(3^2\), and the last is 36, which is \(6^2\).
This implies that our index, often denoted as \(n\), must start at 3 and continue through 6.
Think of the index \(n\) as a pointer that tells the summation where to start and where to conclude.
Therefore, the index range of our sequence of perfect squares is from 3 to 6. By clearly defining this range, we ensure that our summation covers the entire set of numbers provided. This precise demarcation of start and end points is what allows the summation notation to accurately capture the series.

Perfect Squares

Perfect squares are numbers that can be expressed as an integer multiplied by itself. They are fundamental to recognizing more complex number patterns and are often utilized in simplification techniques in algebra and calculus.
In this exercise, understanding perfect squares is essential because each number in our sequence - 9, 16, 25, and 36 - is derived by squaring whole numbers: 3, 4, 5, and 6 respectively.

• Perfect squares build upon one another in a predictable way, which helps when reducing numbers to a simpler form and when setting summation as in our example.
• They often appear in various mathematical problems, especially those dealing with area (since area calculations frequently involve squares) or as part of larger numerical patterns.

Recognizing and working with perfect squares allows for a deeper grasp of mathematical relationships and enhances problem-solving efficiency, along with significantly simplifying much of the work involved in expressions and calculations.