Problem 23 - Algebra 2 and Trigonometry

Question

In \(19-24 :\) a. Write each arithmetic series as the sum of terms. b. Find the sum. $$ \sum_{n=1}^{25}-2 n $$

Step-by-Step Solution

Answer

The sum of the series is -650.

Step 1: Understanding the Series

The given series is \( \sum_{n=1}^{25} -2n \). This means we are summing each term \(-2n\) starting at \(n = 1\) and ending at \(n = 25\).

Step 2: Writing the Series as Sum of Terms

We write out the series by calculating each term from \(n=1\) to \(n=25\):\(-2(1), -2(2), -2(3), \ldots, -2(25)\). This can be written explicitly as \(-2(1) + -2(2) + -2(3) + \ldots + -2(25)\).

Step 3: Simplifying Each Term

Simplify each term by performing the multiplication with \(-2\): \(-2, -4, -6, \ldots, -50\).

Step 4: Calculating the Total Sum

Observe that the series is arithmetic with the first term \(-2\), the common difference \(-2\), and the last term \(-2(25) = -50\). The sum of an arithmetic series is given by:\[ S_n = \frac{n}{2} (a + l) \] where \(a\) is the first term, \(l\) is the last term, and \(n\) is the number of terms.

Step 5: Applying the Formula for Sum

For this series, \(a = -2\), \(l = -50\), and \(n = 25\). Substitute these values into the sum formula:\[ S_{25} = \frac{25}{2}(-2 + (-50)) \] which simplifies to \[ S_{25} = \frac{25}{2} \times (-52) \].

Step 6: Final Calculation

Calculate the sum: \( \frac{25}{2} \times -52 = 25 \times -26 = -650 \). Thus, the sum of the series is \(-650\).

Key Concepts

Sum of Arithmetic Series

When we talk about the sum of an arithmetic series, we really mean adding up the terms of a sequence where each term increases by a constant amount, called the common difference. If you're given a sequence where terms shift consistently, this concept helps us quickly find the complete sum. To make life easier, use the formula:

• \[ S_n = \frac{n}{2} (a + l) \]

Here, \(S_n\) stands for the total sum you aim to find, \(n\) is the count of terms, \(a\) is your starting term, and \(l\) is the last term of this sequence.
For example, in the series \( \sum_{n=1}^{25} -2n \), the first term \(a\) is \(-2\), the last term \(l\) hits \(-50\), and you've got 25 terms. Plug these values into our handy formula, and voilà! You effortlessly find the total sum: \(-650\). Understanding this helps you tackle other problems involving arithmetic sums with ease.

Sequence Calculation

Calculating the terms in a sequence involves understanding how each term is related. In an arithmetic sequence, every term is generated by adding the common difference to the previous one. It could be positive or negative.
Let's dig into this using our sequence from the exercise. If you start with \(-2(1) = -2\), each following term increases in steps of \(-2\), leading to \(-4, -6, -8, \ldots, -50\).
When you have 25 terms, expressing them individually can be daunting, but focusing on the pattern makes it manageable.

• Identify the first term: here, it's \(-2\).
• Note the common difference: in this series, it's \(-2\).
• Simplify, or develop each term consecutively.

This systematic approach ensures that no term is missed, keeping everything neat and tidy.

Arithmetic Sequence Formula

The arithmetic sequence follows a simple idea: sequence terms change regularly by adding or subtracting the common difference. This consistency can be captured mathematically using the formula:

• \[ a_n = a + (n-1) imes d \]

Here, \(a_n\) represents any term within the sequence, \(a\) is the first term, \(n\) tells us which term we are interested in, and \(d\) is the common difference.
When you apply this formula, determining specific terms in the sequence becomes easy. Our example starts at \(-2\) and each following term reduces by \(-2\). Calculating, say, the 10th term? Simply plug into the formula:

• \[ a_{10} = -2 + (10-1) \times (-2) = -2 + (-18) = -20 \]

Using the arithmetic sequence formula makes calculating specific terms a breeze. For bigger problems, this technique provides a simple but robust tool for tackling sequences.