Problem 55 - Algebra for College Students

Question

Find the sum of the first 30 terms of the arithmetic sequence with the general term \(a_{n}=5 n-4 . \quad 2205\)

Step-by-Step Solution

Answer

The sum of the first 30 terms of the sequence is 2205.

Step 1: Understanding the Arithmetic Sequence

The general term of an arithmetic sequence is given as \(a_n = 5n - 4\). We must find the sum of the first 30 terms. Let's denote the first term \(a_1\) and the nth term \(a_n\) for ease of calculations.

Step 2: Calculating the First Term

Substitute \(n = 1\) into the general term formula to find the first term. This gives us: \[a_1 = 5(1) - 4 = 1\]. This means the first term of the sequence is \(1\).

Step 3: Calculating the 30th Term

Substitute \(n = 30\) into the general term formula to find the 30th term. This gives: \[a_{30} = 5(30) - 4 = 146\]. Therefore, the 30th term of the sequence is \(146\).

Step 4: Using the Sum Formula for Arithmetic Sequences

The sum of the first \(n\) terms of an arithmetic sequence \(S_n\) is given by the formula \[S_n = \frac{n}{2} (a_1 + a_n)\]. Here, \(n = 30\), \(a_1 = 1\), and \(a_{30} = 146\).

Step 5: Calculating the Sum

Plug the values into the sum formula: \[S_{30} = \frac{30}{2} (1 + 146) = 15 \times 147\].

Step 6: Final Calculation

Calculate \(15 \times 147\), which equals \(2205\). Thus, the sum of the first 30 terms is \(2205\).

Key Concepts

Sum of Arithmetic Series

When dealing with arithmetic sequences, the sum of the series is an important concept that involves adding up all the terms in the sequence. An arithmetic series refers to the sum of the terms in an arithmetic sequence, which is a collection of numbers with a constant difference between consecutive terms. To find the sum of the first "n" terms of an arithmetic sequence, we use the formula:\[ S_n = \frac{n}{2} (a_1 + a_n) \]where:

• \( S_n \) is the sum of the first "n" terms,
• \( n \) is the number of terms to be added up,
• \( a_1 \) is the first term of the sequence,
• \( a_n \) is the nth term of the sequence.

This formula arises from the idea that by pairing the terms from the beginning and end of the sequence, each pair will give the same sum. This method simplifies the calculation of large sums in arithmetic sequences.

General Term Formula

The general term formula for an arithmetic sequence helps you find any term in the sequence. An arithmetic sequence has a specific pattern where each term after the first is the result of adding a constant value, known as the common difference, to the previous term.For any given arithmetic sequence, the formula for the nth term is:\[ a_n = a_1 + (n-1)d \]where:

• \( a_n \) is the nth term you want to find,
• \( a_1 \) is the first term of the sequence,
• \( n \) is the term number,
• \( d \) is the common difference between consecutive terms.

In our example, the general term is given as \( a_n = 5n - 4 \). This expression allows for direct calculation of any term in the sequence without needing to calculate all the previous terms.

Arithmetic Sequence Formula

Arithmetic sequences follow a simple structure where each term is derived by adding a fixed number, known as the common difference, to the preceding term. The arithmetic sequence formula allows us to find subsequent terms or validate the structure of the sequence.The formula used to define the general expression for an arithmetic sequence is:\[ a_n = a_1 + (n-1)d \]This formula shows that with the first term \( a_1 \) and the common difference \( d \), you can find any term in the sequence by plugging in \( n \), the position of the term in sequence.In the problem, the general term provided, \( a_n = 5n - 4 \), reflects an arithmetic sequence where the coefficient of \( n \) (which is 5) represents the common difference. This means each term increases by 5 from the previous one.

First and nth Term Calculation

Finding the first and nth terms are fundamental in solving arithmetic sequence problems because they are used to calculate the sum of the series and understand its pattern.To find the first term \( a_1 \), substitute \( n = 1 \) into the general term formula. In the case of the problem, substituting gives:\[ a_1 = 5(1) - 4 = 1 \]This shows the sequence starts with \( 1 \).For the nth term, substitute the desired term position into the general term formula. For instance, to find the 30th term \( a_{30} \), substitute \( n = 30 \):\[ a_{30} = 5(30) - 4 = 146 \]This calculation reveals that the 30th term is \( 146 \). Knowing both the first and nth terms allows you to use the sum formula efficiently to find the sum of a sequence's terms.