Problem 49 - Calculus Single Variable

Question

Calculate the sum \(S=\sum_{j=1}^{N}(2 j)^{2}\) of the first \(N\) even positive square integers. Subtract your value of \(S\) from the value of \(\sum_{j=1}^{2 N} j^{2}\) to calculate the sum \(\sum_{j=1}^{N}(2 j-1)^{2}\) of the first \(N\) odd positive square integers

Step-by-Step Solution

Answer

The sum of the first \( N \) odd positive square integers is \( \frac{N(4N^2-1)}{3} \).

Step 1: Understand the Problem

We need to calculate two sums. The first is the sum of squares of the first \( N \) even numbers, denoted as \( S = \sum_{j=1}^{N}(2j)^2 \). The second sum is the sum of squares of the first \( N \) odd numbers which can be found by subtracting \( S \) from the sum of squares of the first \( 2N \) integers \( \sum_{j=1}^{2N} j^2 \).

Step 2: Expression for the First Sum

The first sum is \( S = \sum_{j=1}^{N} (2j)^{2} = \sum_{j=1}^{N} 4j^{2} \). This simplifies to \( 4 \sum_{j=1}^{N} j^{2} \). We use the formula for the sum of squares: \( \sum_{j=1}^{N} j^2 = \frac{N(N+1)(2N+1)}{6} \).

Step 3: Calculate \( S \)

Now calculate \( S = 4 \cdot \frac{N(N+1)(2N+1)}{6} = \frac{2N(N+1)(2N+1)}{3} \).

Step 4: Calculate the Sum of Squares of the First \( 2N \) Integers

The sum of squares of the first \( 2N \) integers is \( \sum_{j=1}^{2N} j^2 = \frac{2N(2N+1)(4N+1)}{6} \).

Step 5: Calculate the Sum of Squares of the First \( N \) Odd Integers

To find the sum of squares of the first \( N \) odd integers, subtract \( S \) from the sum of squares of the first \( 2N \) integers, i.e., \( \sum_{j=1}^{N}(2j-1)^{2} = \frac{2N(2N+1)(4N+1)}{6} - \frac{2N(N+1)(2N+1)}{3} \).

Step 6: Simplify the Final Expression

Simplify the expression to get the sum of squares of the first \( N \) odd integers. Use common denominators and algebraic expansion to simplify the complete expression. The simplified expression results in: \( N(4N^2-1)/3 \).

Key Concepts

Understanding Even Integers

Even integers are numbers that can be divided by 2 without leaving a remainder. They appear regularly in arithmetic sequences and are critical when dealing with the sum of squares problems.
For example, numbers like 2, 4, and 6 are even integers. In our exercise, we specifically focus on the even integers formulated as \(2j\), where \(j\) is a positive integer.

• This formulation allows us to systematically work through problems involving even numbers.
• Each even integer can be represented as double an integer, which makes calculations involving sums and products more systematic.
• Understanding even numbers is pivotal in calculating the sum of their squares.

Once you grasp the nature of even integers, tackling their algebraic sums becomes straightforward.

Exploring Odd Integers

Odd integers are numbers that cannot be evenly divided by 2; they always leave a remainder of 1. Similar to even integers, they follow a predictable pattern that is beneficial when solving algebraic expressions.
Typical examples include 1, 3, and 5. In our exercise, we describe odd numbers with the expression \(2j - 1\), again with \(j\) being a positive integer.

• This specific form creates a clear distinction from even numbers.
• Using \(2j - 1\) emphasizes their regular interval and helps in systematic calculations.
• Understanding odd numbers helps when calculating expressions related to their squares.

This foundational knowledge allows you to confidently approach mathematical problems involving odd integers.

Decoding Algebraic Expressions

Algebraic expressions are mathematical phrases involving numbers, variables, and operations. Our focus in this exercise was to strategically use these expressions to find sums of squares.
We transformed known series into simple algebraic forms to handle complexity efficiently.

• The first sum, \( \sum_{j=1}^{N}(2j)^2 \), was simplified to \( 4 \sum_{j=1}^{N} j^2 \).
• This transformation uses the property \((2j)^2 = 4j^2\) to ease calculations.
• Such algebraic manipulations are pivotal in simplifying and solving complex summation problems.

Understanding how to create and simplify algebraic expressions is key to succeeding in mathematical tasks.

Conducting a Mathematical Proof

Mathematical proofs are logical arguments demonstrating the truth of a mathematical statement. In this exercise, proofs ensured we accurately calculated complex sums and verified our solutions.
We started by expressing the sums using formulas and then subtracting to derive the desired results.

• Each step relied on mathematical logic and known formulas for sums of squares.
• By structuring the calculation as a proof, we ensured the results were reliable and verified.
• Proof is about showing the correctness of an expression beyond merely calculating a value.

Understanding and conducting proofs allow us to validate work methodically, reducing errors and increasing confidence in our solutions.