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Chapter 19: Problem 4

Find the sum of the first 40 positive integers.

Short Answer

Expert verified
Answer: The sum of the first 40 positive integers is 820.

Step by step solution

01

Identify the formula

We will use the sum of arithmetic series formula, which is S = (n * (n + 1))/2. In this case, n=40 as it's the sum of the first 40 positive integers.
02

Substitute the values in the formula

Now, substitute the value of n=40 in the formula S = (n * (n + 1))/2. S = (40 * (40 + 1))/2
03

Simplify the expression

Now, simplify the expression to find the sum. S = (40 * (41))/2
04

Perform the calculations

Perform the calculations to find the sum. S = (40 * 41) / 2 S = 1640 / 2
05

Find the sum

Finally, find the sum S = 820. Hence, the sum of the first 40 positive integers is 820.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Integers
The sum of integers is a foundational concept in mathematics that helps us calculate the total of a series of numbers. In practical terms, when asked to find the sum of the first 40 positive integers, it means we want to add all the numbers from 1 to 40. Adding each number individually is time-consuming, especially as the list grows longer. To simplify this, mathematicians have developed formulas, such as the sum formula for arithmetic series, which allow us to perform such calculations quickly and accurately. Calculating the sum of integers efficiently is crucial in various applications, from simple math problems to more complex scientific calculations.
Arithmetic Sequence
An arithmetic sequence is a series of numbers with a constant difference between consecutive terms. The first term, known as the common difference (usually denoted as \(d\)), is added to each number to get the next in the sequence. For example, in the sequence 1, 2, 3, ..., 40, each number increases by 1.The formula for any term in an arithmetic sequence is:
  • \(a_n = a_1 + (n-1) \cdot d\)
Where:
  • \(a_n\) is the \(n^{th}\) term.
  • \(a_1\) is the first term.
  • \(d\) is the common difference. For this series, \(d = 1\).
Understanding arithmetic sequences is crucial as they often underlie problems involving sums of numbers, such as the one we're discussing.
Sum Formula
The sum formula for an arithmetic series is a quick and efficient method to find the total of a sequence of numbers. For the sum of the first \(n\) integers, we use the formula:
  • \(S = \frac{n(n+1)}{2}\)
This formula originates from clever mathematical insight, historically attributed to the mathematician Gauss. When asked to sum a long list of numbers, positioning the sequence to make pairs that have the same sum (e.g., \(1+40\), \(2+39\), \dots) simplifies the process and leads us to this formula.In our exercise, using \(n = 40\), we found the sum through:
  • \(S = \frac{40(40+1)}{2}\)
Which simplifies to the total sum of 820, demonstrating the power and utility of the sum formula in arithmetic sequences.

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Most popular questions from this chapter

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