Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, find the indicated sum. \(\sum_{a=1}^{14} a\)

Step-by-Step Solution

Answer

The sum is 105.

Step 1: Understanding the Sum

The problem asks us to find the sum of all integers from 1 to 14. This is denoted as \(\sum_{a=1}^{14} a\). It means you need to add up all the integers starting from \(a = 1\) to \(a = 14\).

Step 2: Apply Formula for Sum of an Arithmetic Series

The sum of the first \(n\) natural numbers is given by the formula \(\frac{n(n+1)}{2}\). Here, \(n = 14\), so we substitute 14 into the formula.

Step 3: Substitute Values into the Formula

Use the formula from Step 2: \(\frac{14(14+1)}{2}\). Calculate \(14 + 1 = 15\), and then \(14 \times 15 = 210\).

Step 4: Divide to Find the Sum

Now divide 210 by 2: \(\frac{210}{2} = 105\). This is the sum of the numbers from 1 to 14.

Key Concepts

Natural Numbers

Natural numbers are the set of positive integers starting from 1 and continuing infinitely: 1, 2, 3, 4, and so on. These numbers are the building blocks for arithmetic and are often referred to as counting numbers as they naturally arise when counting objects. In arithmetic, natural numbers are crucial in understanding sequences and series, especially when calculating sums.

When problems involve summing these numbers, they usually range between a specific starting point and an endpoint. For example, in the exercise given, the natural numbers from 1 to 14 are summed. Understanding natural numbers is essential as it forms the basis of more complex operations in arithmetic, such as addition, subtraction, and especially in arithmetic sequences.

Arithmetic Sequence

An arithmetic sequence is a sequence of numbers in which each term after the first is obtained by adding a fixed number, called the common difference, to the previous term. This means it is a series of numbers with a constant difference between consecutive terms.

For example, in the sequence 2, 4, 6, 8, the common difference is 2. Each term is 2 more than the previous one.

• Initial term: the first number in the sequence.
• Common difference: the fixed amount added to each term to get the next one.

In the exercise example, the arithmetic sequence comprises the natural numbers from 1 to 14 with a common difference of 1. Each number is simply 1 more than the previous number.

Series Sum Formula

To find the sum of an arithmetic sequence, we use the series sum formula. This formula efficiently calculates the total of all terms in the sequence without needing to add each term individually.

For a sequence with first term \(a\), last term \(l\), and number of terms \(n\), the sum \(S\) can be calculated with the formula: \[S = \frac{n}{2} \times (a + l)\]For the sum of the first \(n\) natural numbers, a simplified version of this formula is often used:\[S = \frac{n(n+1)}{2}\]This formula arises since the first term is 1, the last term is \(n\), and the sequence increases by 1 each time.

• Plug in the endpoint number as \(n\).
• Calculate \(n + 1\).
• Multiply \(n\) by \(n+1\).
• Divide the result by 2 for the sum.

Using the series sum formula helps solve problems like finding the sum of numbers from 1 to 14 quickly and accurately, as demonstrated in the exercise.