Problem 32 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) for each arithmetic sequence. $$S_{28}=2926, a_{28}=199$$

Step-by-Step Solution

Answer

The first term, \( a_1 \), is 10.

Step 1: Identify the formula for the sum of an arithmetic sequence

The sum of the first \( n \) terms of an arithmetic sequence is given by the formula: \[ S_n = \frac{n}{2}(a_1 + a_n) \] where \( S_n \) is the sum of the first \( n \) terms, \( n \) is the number of terms, \( a_1 \) is the first term, and \( a_n \) is the \( n \)th term.

Step 2: Substitute given values into the sum formula

Given \( S_{28} = 2926 \), \( a_{28} = 199 \), and \( n = 28 \). Substitute these values into the sum formula: \[ 2926 = \frac{28}{2}(a_1 + 199) \] Simplifying gives: \[ 2926 = 14(a_1 + 199) \]

Step 3: Solve for \( a_1 \)

First, divide both sides by 14 to isolate \( a_1 + 199 \): \[ \frac{2926}{14} = a_1 + 199 \] Calculating gives: \[ 209 = a_1 + 199 \] Subtract 199 from both sides to solve for \( a_1 \): \[ 209 - 199 = a_1 \] \[ a_1 = 10 \]

Key Concepts

sum of sequences

Understanding the sum of sequences is essential in solving various problems within arithmetic sequences. An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. In this context, the sum of an arithmetic sequence involves adding all the terms up to a certain point.
To find the sum of any arithmetic sequence, the following formula is used: \[ S_n = \frac{n}{2}(a_1 + a_n) \] This tool provides the sum \( S_n \) of the first \( n \) terms. Here's a breakdown of the components:

• \( n \) - Total number of terms you're summing up.
• \( a_1 \) - The first term in the sequence.
• \( a_n \) - The nth or last term in the sequence up to which you're calculating the sum.

By substituting the known values into this formula, solving problems related to sums of sequences becomes straightforward. Just plug in the terms, simplify, and calculate to find your desired sum.

first term calculation

The first term calculation is often a critical step in understanding an arithmetic sequence. It refers to determining the initial number in the sequence when given certain parameters, like the sum of a series and the last term of the sequence. To find this first term, the sum formula of the sequence is invaluable.

For example, given:

• Total sum \( S_{28} = 2926 \)
• Last term \( a_{28} = 199 \)

Substitute these into the sum formula: \[ 2926 = \frac{28}{2}(a_1 + 199) \]
This will allow you to rearrange the equation to isolate \( a_1 \) and eventually solve for it. First, calculate \( 2926 \div 14 \) to simplify:
\[ 209 = a_1 + 199 \]Finally, to find \( a_1 \), subtract 199 from 209 resulting in \( a_1 = 10 \). This straightforward calculation ensures you accurately determine the first part of the sequence.

sequence formulas

Sequence formulas are essential tools for anyone dealing with arithmetic sequences. They help to decipher unknowns within a sequence by providing a mathematical framework to work with.

• Sum of the Sequence Formula: \( S_n = \frac{n}{2}(a_1 + a_n) \), assists in determining the total sum of terms in a sequence when the first and last terms, as well as the number of terms, are known.
• General Term Formula: In case you need it for extending sequences or calculations, recall that the \( n \)th term \( a_n \) can also be found using \( a_n = a_1 + (n-1) \cdot d \), where \( d \) is the common difference.

These formulas are like the keys to encrypted sequences. They translate known values into insightful solutions, helping to uncover other aspects of the arithmetic sequence, such as intermediate terms or specific term values.