Problem 30 - College Algebra

Question

Find the indicated sum. $$ \sum_{a=1}^{14} a $$

Step-by-Step Solution

Answer

The sum is 105.

Step 1: Understand the Problem

You are asked to find the sum of a series of numbers starting from 1 and ending at 14. This can be written as the series \(1 + 2 + 3 + \, \ldots \, + 14\).

Step 2: Identify the Formula

The sum of the first \(n\) natural numbers can be calculated using the formula \(S = \frac{n(n+1)}{2}\). Here, \(n\) is the last number in the series, which is 14 in this case.

Step 3: Substitute Values

Substitute \(n = 14\) into the formula:\[S = \frac{14(14+1)}{2} = \frac{14 \times 15}{2}\]

Step 4: Perform Calculations

Calculate the product and the division to find the sum:\[S = \frac{210}{2} = 105\]

Step 5: Verify the Calculation

You can sum the numbers manually as a check: \(1 + 2 + 3 + \ldots + 14\) adds up to 105, confirming our formula calculation.

Key Concepts

Arithmetic Series

An arithmetic series is a sequence of numbers in which each term increases by a constant value from the previous one. This constant value is known as the "common difference." In our example, the series starts at 1 and increases by 1 each time till it reaches 14. This means it's an arithmetic series with a common difference of 1.
Understanding the properties of an arithmetic series helps in calculating the sum efficiently without manually adding each term. The sum of an arithmetic series can be quickly determined using a formula rather than adding each number one by one. This saves time and reduces errors, especially when dealing with larger series.

• The first term is often represented as \(a_1\).
• The last term here is 14, while the first is 1.
• Common difference \(d\) here is 1.

Understanding these components is key to identifying and summing an arithmetic series.

Summation Formula

The summation formula is a powerful tool used to add up terms in an arithmetic series quickly and efficiently. Instead of adding each term individually, we use the formula:
\[S = \frac{n(n+1)}{2}\]
Here, \(S\) stands for the sum of the series, and \(n\) is the number of terms.
This formula derives from the concept that pairing numbers from both ends of the series yields the same sum. For an arithmetic series from 1 to 14, notice that:

• 1 + 14 = 15
• 2 + 13 = 15
• 3 + 12 = 15

This pattern continues. There are 7 such pairs from 1 to 14, making the multiplication by 15 evident. Understanding the origin of this formula provides a deeper insight into its application and why it works.

Series Calculations

Performing series calculations involves using the summation formula to compute the total sum efficiently. Let's continue with our exercise to see this in action. By using the formula \(S = \frac{n(n+1)}{2}\), and substituting \(n = 14\):
1. Calculate \(14 + 1\): - Result is 15.2. Multiply \(14\) by the result (15): - \(14 \times 15 = 210\)3. Divide 210 by 2: - \(\frac{210}{2} = 105\)This streamlined method ensures accuracy and speed, especially for longer series where manual addition would be laborious. The arithmetic confirms that 105 is indeed the sum of numbers from 1 to 14, as seen in our verification step by manual addition.
Essentially, employing series calculations with a formula empowers students to solve problems efficiently, facilitating a better understanding and mastery over arithmetic sequences.