Problem 49 - Precalculus with Limits

Question

In Exercises 49 - 58, find the sum using the formulas for the sums of powers of integers. \( \sum_{n=1}^{15}n \)

Step-by-Step Solution

Answer

The sum of the first 15 natural numbers using the formula is 120.

Step 1: Identify Formula

Firstly, identify the right formula for the given scenario, which is the sum of the first n natural numbers. The formula is \( \frac{n(n+1)}{2} \). Here, the variable 'n' is equal to 15.

Step 2: Substitute in Formula

Substitute n = 15, into the formula, which gives us \( \frac{15(15 + 1)}{2} \).

Step 3: Perform Calculation

Perform the calculation: \( \frac{15*16}{2} = 120 \).

Key Concepts

Natural Numbers

Natural numbers are the set of positive integers starting from 1. They are the numbers we typically use when counting objects. It is important to note that these numbers do not include zero or any negative numbers. Natural numbers are the simplest form of numbers and are the basis for learning more complex mathematical concepts.
Understanding natural numbers helps in forming a solid foundation for various arithmetic operations like addition, subtraction, multiplication, and division. This is because these numbers are inherently related to counting and practical applications of all basic operations.

• Natural numbers start from 1 and go on infinitely: 1, 2, 3, 4,...
• They do not include fractions or decimals.
• They are the building blocks of other number systems.

Grasping the basics of natural numbers is crucial for comprehending more advanced topics like integer sequences and summation formulas.

Integer Sequences

An integer sequence is an ordered list of numbers where each number is a whole number. In mathematics, a sequence is a set of numbers in a specific order. With integer sequences, all the numbers are integers, which means they can be positive, negative, or zero.
One common example of an integer sequence is the sequence of natural numbers, as seen in the exercise. This sequence is critical when you are examining the sum of a consecutive list of numbers from this set.
Some properties of integer sequences include:

• Each number in the sequence is called a term.
• Sequences can be finite, with a set number of terms, or infinite, continuing endlessly.
• Sequences are often defined by specific rules or formulas that dictate how to generate subsequent terms.

Understanding sequences is vital for delving into the concept of summing these terms using different sum formulas.

Sum Formulas

Sum formulas are mathematical expressions used to find the total of a set sequence of numbers. They allow for quick calculations without the need to add each number individually. For the sequence of natural numbers, there is a specific formula that can find the sum of these numbers up to any integer 'n'.
The formula utilized in the provided exercise is \( \frac{n(n+1)}{2} \). This formula efficiently calculates the sum of the first 'n' natural numbers without manual addition.
Key features of the sum formula for natural numbers:

• The formula is derived from arithmetic series principles.
• It simplifies summation by providing a direct computation method for large sequences.
• Using this formula saves time and ensures accuracy when handling larger data sets.

With the sum formula, additions of large sequences become manageable, providing a useful tool for students and professionals alike. Understanding this formula enhances problem-solving skills and is indispensable in various mathematical applications.