Problem 63 - Contemporary Precalculus

Question

Find the sum of the first 200 positive integers.

Step-by-Step Solution

Answer

Answer: The sum of the first 200 positive integers is 20,100.

Step 1: Identify the number of terms (n)

We are given that the task is to find the sum of the first 200 positive integers. Therefore, n = 200.

Step 2: Apply the formula for the sum of an arithmetic sequence

Now we need to apply the formula for the sum of an arithmetic sequence: Sum = (n * (n + 1)) / 2

Step 3: Substitute the value of n and calculate the sum

Now that we know the value of n, let's substitute it into the formula and find the sum: Sum = (200 * (200 + 1)) / 2

Step 4: Simplify and calculate the final sum

Let's simplify and calculate the final sum: Sum = (200 * 201) / 2 Sum = 40200 / 2 Sum = 20100 So, the sum of the first 200 positive integers is 20,100.

Key Concepts

Sum of Integers

The concept of summing integers, especially a large sequence of numbers, can initially seem daunting. However, when dealing with the sum of consecutive positive integers, there's a straightforward and efficient approach. Consider the example of summing the first 200 integers. Instead of adding each number one by one, we can use a mathematical method. This sum is an arithmetic sequence with a known formula that drastically simplifies the process. Understanding this formula saves time and effort for calculations involving large sets of numbers. Let's discuss this formula next.

Arithmetic Progression Formula

An arithmetic sequence, also known as an arithmetic progression, is a sequence of numbers with a common difference between consecutive terms. This property makes calculating the sum straightforward. For an arithmetic sequence, the sum of the first \( n \) integers from 1 to \( n \), can be calculated using the formula: \[ \text{Sum} = \frac{n(n + 1)}{2} \] Here, \( n \) represents the total number of terms in the sequence. This formula essentially pairs the first and last term in the sequence, continuing this pattern through the list, thereby simplifying the addition process. In our example, \( n \) is 200, so inserting this into the formula gives: \[ \text{Sum} = \frac{200(200 + 1)}{2} \] This method highlights the elegance of arithmetic progressions, allowing for quick calculations without manually adding each number.

Sequence Summation

Summing a sequence isn't just about reaching the final total, but also understanding the mechanics behind it. Sequence summation is the process of adding a sequence of numbers. In arithmetic sequences, this is often accomplished using specific formulas tailored to the sequence's properties. In practical terms, sequence summation involves identifying the series type and applying the relevant formula. When dealing with simple sequences, like consecutive integers, it's crucial to leverage formulas to avoid error-prone and time-intensive manual addition. By plugging in numbers into the arithmetic progression formula, one can easily compute large sums, such as those found in our problem of summing 200 integers, thus demonstrating the power and efficiency of sequence summation in mathematics.