Problem 25 - Algebra for College Students

Question

Find the sum of the first 95 terms of the sequence where $$ a_{n}=7 n+1 . \quad 32,015 $$

Step-by-Step Solution

Answer

The sum of the first 95 terms of the sequence is 32,015.

Step 1: Identify the Sequence

The given sequence is an arithmetic sequence with the formula for the general term given as \( a_n = 7n + 1 \). This means the first term \( a_1 \) is \( 8 \), calculated by substituting \( n=1 \) into the formula, and this sequence increases by a common difference of \( 7 \).

Step 2: Determine Sequence Parameters

With \( a_1 = 8 \) and \( a_n = 7n + 1 \), calculate the 95th term \( a_{95} \). Substitute \( n = 95 \) into the general term: \( a_{95} = 7 imes 95 + 1 = 666 \).

Step 3: Use Arithmetic Series Formula

The sum of an arithmetic sequence can be calculated using the formula \( S_n = \frac{n}{2} (a_1 + a_n) \), where \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the last term. Here, \( n = 95 \), \( a_1 = 8 \), and \( a_{95} = 666 \).

Step 4: Substitute Values

Substitute the known values into the formula: \( S_{95} = \frac{95}{2} (8 + 666) \). Simplify inside the brackets: \( 8 + 666 = 674 \), giving \( S_{95} = \frac{95}{2} \times 674 \).

Step 5: Calculate the Sum

Simplify to find \( S_{95} = 47.5 \times 674 = 32015 \). Therefore, the sum of the first 95 terms is 32,015.

Key Concepts

Understanding Arithmetic Series

An arithmetic series is essentially the sum of elements from an arithmetic sequence. An arithmetic sequence itself is a list of numbers with a unique property: the difference between consecutive terms is constant. This constant is known as the "common difference." For instance, consider the sequence: 8, 15, 22, 29, ... Here, each term increases by 7, which is our common difference.

In this type of series, the goal is often to find the sum of a certain number of terms. Understanding how these terms relate is key to applying the series formula effectively. Recognizing the pattern and verifying the common difference helps predict not only the series behavior but also calculate sums without needing to list each term individually.

Deriving Sequence Formulas

Sequence formulas are our tools for efficiently describing each term in a sequence without manually counting. For an arithmetic sequence, the formula for any term is based on its position in the sequence and the common difference. The general formula looks like this: \( a_n = a_1 + (n-1) \cdot d \), where:

• \( a_n \) is the \( n \)-th term.
• \( a_1 \) is the first term.
• \( n \) is the term number.
• \( d \) is the common difference.

Knowing this formula allows you to quickly find any term within the sequence, which is critical when you're asked to find the sum of several terms. Using the formula, verify the terms easily: substitute into \( a_n = 7n + 1 \) to find any term like the 95th term in our sequence example, giving \( a_{95} = 666 \).

Calculating the Sum of Sequences

Calculating the sum of an arithmetic sequence involves combining the first term, the last term you want to consider, and the total number of terms. The standard formula for this sum is:\[ S_n = \frac{n}{2} \times (a_1 + a_n) \]where \( S_n \) is the sum of the first \( n \) terms, \( a_1 \) is the first term, and \( a_n \) is the \( n \)-th term.

This formula works because it averages the first and last terms but considers all terms. Let's comprehend this with our example with the sequence formula \( a_n = 7n + 1 \):

• For 95 terms, \( a_1 = 8 \) and \( a_{95} = 666 \).
• Substitute into sum formula to get \( S_{95} = \frac{95}{2} \times (8 + 666) \).
• Simplify calculations: \( S_{95} = 47.5 \times 674 = 32015 \).

This result, 32,015, tells us the total when combining the first 95 terms of the sequence.