Problem 57 - Algebra for College Students

Question

Find the sum of the first 25 terms of the arithmetic sequence with the general term \(a_{n}=-4 n-1\). \(-1325\)

Step-by-Step Solution

Answer

The sum of the first 25 terms is -1325.

Step 1: Understanding the Arithmetic Sequence

The general term of an arithmetic sequence is given as \(a_n = -4n - 1\). This formula helps us find any term in the sequence. Since we need the first 25 terms, our \(n\) will range from 1 to 25.

Step 2: Finding the First Term (\(a_1\))

Substitute \(n=1\) into the general formula to find the first term. \(a_1 = -4(1) - 1 = -4 - 1 = -5\). The first term of the sequence is \(-5\).

Step 3: Finding the 25th Term (\(a_{25}\))

Substitute \(n=25\) into the general formula to find the 25th term. \(a_{25} = -4(25) - 1 = -100 - 1 = -101\). The 25th term of the sequence is \(-101\).

Step 4: Using the Sum Formula for Arithmetic Sequence

The sum of the first \(n\) terms of an arithmetic sequence is given by the formula \(S_n = \frac{n}{2} (a_1 + a_n)\). Here, \(n = 25\), \(a_1 = -5\), and \(a_{25} = -101\).

Step 5: Substituting to Find Sum

Use \(S_{25} = \frac{25}{2}(-5 - 101)\). First, calculate \(-5 - 101 = -106\), then substitute: \(S_{25} = \frac{25}{2} imes -106\).

Step 6: Calculating the Final Sum

Simplify \(S_{25} = \frac{25}{2} imes -106\). First, calculate \(-106 = -106\) and then multiply: \(S_{25} = 25 imes -53 = -1325\). Therefore, the sum of the first 25 terms is \(-1325\).

Key Concepts

Sum of an Arithmetic Sequence

Finding the sum of an arithmetic sequence involves adding up all the terms in the sequence. When an arithmetic sequence has an identified first term and a constant difference between each consecutive term, a specific formula can be used to simplify this process:

• The sum of the first n terms of an arithmetic sequence is calculated using the formula: \[ S_n = \frac{n}{2} (a_1 + a_n) \]

Here, \(S_n\) represents the sum of the first \(n\) terms, \(a_1\) is the first term, and \(a_n\) is the nth term. This approach helps simplify what could otherwise be a lengthy addition process, especially for large \(n\). For instance, in the problem given, the sum of the first 25 terms was calculated using this formula, involving the specific first and 25th term, resulting in a total sum of \(-1325\).

General Term Formula

The general term formula allows us to calculate any specific term in an arithmetic sequence without listing all previous terms. In an arithmetic sequence, each term after the first is generated by adding a constant difference to the previous term. The formula is expressed as:

• \[ a_n = a_1 + (n-1) \cdot d \]

Where \(a_n\) is the nth term, \(a_1\) is the first term, \(d\) is the common difference, and \(n\) is the term's position. In the example problem, the general term was provided as \(a_n = -4n - 1\), which allowed the computation of any term directly, like \(a_1 = -5\) and \(a_{25} = -101\), using the corresponding values for \(n\). This formula provides an efficient way to access specific terms without iterating through the entire sequence.

Arithmetic Series

An arithmetic series is essentially the sum of the terms in an arithmetic sequence. Each term in the arithmetic sequence is added sequentially. Knowing the concept of an arithmetic series is critical when solving problems related to summation, like the one given in the exercise.

• Arithmetic series are characterized by having equal intervals between terms, also called the common difference.
• They can be finite or infinite, though infinite series won't be addressed here since the task only concerns a finite number of terms.

The sum of this series is also calculated using the sum formula, illustrating how a potentially long list of additions can be managed with a straightforward calculation.

Finite Series

A finite series refers to a sequence that contains a limited number of terms. Unlike infinite series that go on indefinitely, finite series stop at a specific point, which generally makes them more feasible for computation and analysis.

• In the given exercise, the series is finite because it sums only the first 25 terms of the sequence.
• The formula for the sum of an arithmetic sequence precisely caters to finite series, allowing straightforward computation by considering only a definite number of terms.

Finite series are practical in everyday problems and various applications where a specific range of data points is involved, such as budget forecasting over several months or estimating distances traveled in kilometers.

Sequence Terms Calculation

To calculate specific terms in an arithmetic sequence, you apply the general term formula. Each term is generated through the simple addition of a constant, known as the common difference, to the previous term.

• To find the first term \(a_1\), substitute \(n = 1\) into the general term formula.
• To find the nth term, replace \(n\) with the desired position. This flexibility allows quick determination of any term's value within the sequence without listing all preceding ones.
• In our example, substituting different values of \(n\) into \(a_n = -4n - 1\) provided us with terms like \(a_1 = -5\) and \(a_{25} = -101\).

By understanding this calculation, we can efficiently solve sequence-related problems, ensuring all terms are computed accurately for further analyses or summation.