Problem 50 - Precalculus with Limits

Question

In Exercises 49 - 58, find the sum using the formulas for the sums of powers of integers. \( \sum_{n=1}^{30}n \)

Step-by-Step Solution

Answer

The sum of the first 30 natural numbers is 465.

Step 1: Identify the formula

First, identify the suitable formula for the sum of the first n natural numbers. For this problem, the correct formula is \( \frac{n(n + 1)}{2} \).

Step 2: Apply the formula

Then, substitute the limit 30 into the formula, which is \( \frac{30(30 + 1)}{2} \) or \( \frac{30 \cdot 31}{2} \).

Step 3: Simplify the expression

Finally, compute the multiplication and division operations to get the correct answer. It gives \(\frac{930}{2}\)= 465.

Key Concepts

Natural Numbers

Natural numbers are a set of positive integers which start from 1 and go on indefinitely: 1, 2, 3, and so on.
These are the simplest numbers used for counting and ordering. They do not include zero or any negative values.
When dealing with problems involving natural numbers, we often focus on their sum or sequences.

Natural numbers are integral to many mathematical concepts. Their straightforward nature makes them perfect for beginners to explore arithmetic operations.
When you're summing natural numbers, like in our exercise, you're simply adding them up one by one, or using mathematical formulas to find their collective sum efficiently.

Mathematicians have discovered various patterns and properties in natural numbers. These discoveries led to the development of specific formulas that help simplify these calculations, like the formula for the sum of "n" natural numbers.

Sum of Series

A series is essentially the sum of a sequence. When we talk about the sum of a series, we refer to adding up all the terms within a particular sequence from a starting point to an endpoint.
In the exercise provided, we are summing the first 30 natural numbers.
This can be seen as a series with each term increasing incrementally by one.

The formula used in our exercise is a classic formula for calculating the sum of a series of consecutive natural numbers:

• Formula: \( \frac{n(n + 1)}{2} \)

This formula calculates the sum directly without needing to add each number one at a time.
Here "n" stands for the last number of the sequence, which is 30 in our problem.

The beauty of using such formulas is the convenience and time they save, especially for long lists of numbers. They simplify complex arithmetic into quick calculations, making it manageable and accurate to compute large sums.

Algebraic Expressions

Algebraic expressions involve variables and constants that are combined using operations like addition, subtraction, multiplication, and division.
They are the building blocks of algebra and allow us to perform calculations that might be impossible to solve otherwise.
In our problem, the formula \( \frac{n(n + 1)}{2} \) is an algebraic expression.

• This particular expression calculates the sum of the first "n" natural numbers efficiently.
• In it, "n" is a variable representing the highest natural number to include in the sum.
• The expression leverages multiplication and division to aggregate the sum quickly.

Using algebraic expressions helps to generalize mathematical concepts and solve problems without direct computation of every term.
This makes them incredibly useful for mathematicians and students alike.