Problem 112 - Precalculus with Limits

Question

(a) Compute the following sums of consecutive positive odd integers. \( 1 + 3 = \) \( 1 + 3 + 5 = \) \( 1 + 3 + 5 + 7 = \) \( 1 + 3 + 5 + 7 + 9 = \) \( 1 + 3 + 5 + 7 + 9 + 11 = \) (b) Use the sums in part (a) to make a conjecture about the sums of consecutive positive odd integers. Check your conjecture for the sum \( 1 + 3 + 5 + 7 + 9 + 11 + 13 = \) (c) Verify your conjecture algebraically.

Step-by-Step Solution

Answer

The sums of consecutive positive odd integers starting from 1 are \(2^2\), \(3^2\), \(4^2\), \(5^2\), \(6^2\). It suggests that the sum of n consecutive, positive odd integers starting from 1 is equal to \(n^2\). This is checked for the sum of seven consecutive odd integers and confirmed algebraically as well.

Step 1: Calculation

Calculating the sums goes as follows: \n 1 + 3 = \(4\).\n 1 + 3 + 5 = \(9\).\n 1 + 3 + 5 + 7 = \(16\).\n 1 + 3 + 5 + 7 + 9 = \(25\).\n 1 + 3 + 5 + 7 + 9 + 11 = \(36\).

Step 2: Conjecture

Observing the results from step 1, it is seen that all the sums are perfect squares: \(4 = 2^2\), \(9 = 3^2\), \(16 = 4^2\), \(25 = 5^2\) and \(36 = 6^2\). This suggests that the sum of n consecutive, positive odd integers starting from 1 is equal to \(n^2\).

Step 3: Verification of Conjecture

To validate the conjecture, we check it for the sum of \(1 + 3 + 5 + 7 + 9 + 11 + 13\), which is 7 consecutive positive odd integers. The sum is \(49\) which is \(7^2\), confirming the conjecture.

Step 4: Algebraic Verification

To verify the conjecture algebraically, consider the sum of the first n positive odd integers, which can be written as follows: \(\sum_{i=1}^{n}(2i-1) = 2\sum_{i=1}^{n}i - n = 2\frac{n*(n+1)}{2} - n = n^2 + n - n = n^2\). This proves the conjecture algebraically.

Key Concepts

Odd Numbers

Odd numbers are integers that cannot be divided evenly by 2. They are always one unit away from a neighboring even number. For example, 1, 3, 5, 7, and 9 are all odd numbers. Here are some important characteristics of odd numbers:

• They are of the form \(2k + 1\), where \(k\) is an integer.
• When you add two odd numbers, you'll always get an even number. For instance, 1 + 3 = 4, which is even.
• When you multiply an odd number by an odd number, the result is always odd, such as 3 × 5 = 15.

Odd numbers play a significant role in various mathematical patterns and help in forming conjectures, such as the one describing the sum of consecutive odd integers discussed in this topic.

Perfect Squares

A perfect square is a number that can be expressed as the product of an integer with itself. In other words, if \(x\) is a perfect square, it can be represented as \(x = n^2\), where \(n\) is an integer. Understanding perfect squares is crucial because of their predictable properties.

• Perfect squares include 1, 4, 9, 16, and so on.
• The difference between consecutive perfect squares is always odd, like \(4 - 1 = 3\) and \(9 - 4 = 5\).
• They are often results of the sum of consecutive odd numbers, which becomes evident in these exercises.

In this context, we learn that the sum of the first \(n\) consecutive odd integers results in a perfect square. This relationship adds an intuitive layer for verifying number patterns algebraically and visually.

Algebraic Verification

Algebraic verification provides a more rigorous way to confirm mathematical conjectures. In this scenario, we use it to prove that the sum of the first \(n\) consecutive odd numbers equals \(n^2\). This process helps in making informed and confident conclusions.
Here's a step-by-step explanation of how you can verify this:

• Consider the sum of the first \(n\) consecutive odd numbers: \(1, 3, 5, ..., 2n-1\).
• The sum can be expressed as: \(\sum_{i=1}^{n}(2i-1)\), which simplifies to \(2\sum_{i=1}^{n}i - n\).
• This simplifies using the formula for the sum of the first \(n\) integers: \(2\frac{n(n+1)}{2} - n\), which further breaks down to \(n^2 + n - n\) or simply \(n^2\).

Algebraic verification is a powerful tool that confirms the plausibility of numerical patterns, ensuring that what we observe with numbers holds true as a general mathematical rule. It reassures us that our conjecture about sums of consecutive odd numbers always resulting in perfect squares is indeed valid.