Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{25}=650, a_{25}=62$$

Step-by-Step Solution

Answer

The first term \(a_1\) is \(-10\) and the common difference \(d\) is \(3\).

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

The formula for the sum of the first \( n \) terms of an arithmetic sequence is \( S_n = \frac{n}{2} (a_1 + a_n) \). Given \( S_{25} = 650 \) and \( a_{25} = 62 \), substitute these values into the formula to find the equation \( 650 = \frac{25}{2} (a_1 + 62) \).

Step 2: Simplify the equation for the sum

Simplify the equation from Step 1: \( 650 = \frac{25}{2} (a_1 + 62) \). Multiply both sides by 2 to get \( 1300 = 25(a_1 + 62) \). Divide by 25 to obtain \( a_1 + 62 = 52 \).

Step 3: Solve for the first term \(a_1\)

Rearrange the equation from Step 2: \( a_1 + 62 = 52 \) to solve for \( a_1 \). Subtract 62 from both sides to get \( a_1 = -10 \).

Step 4: Use the nth-term formula for arithmetic sequences

The formula for the \( n \)-th term of an arithmetic sequence is \( a_n = a_1 + (n-1) \cdot d \). Given \( a_{25} = 62 \) and \( a_1 = -10 \), substitute these into the formula: \( 62 = -10 + 24d \).

Step 5: Solve for the common difference \(d\)

Rearrange the equation from Step 4: \( 62 = -10 + 24d \). Add 10 to both sides to get \( 72 = 24d \). Divide by 24 to find \( d = 3 \).

Key Concepts

Sum Formula for Sequences

When dealing with arithmetic sequences, one important aspect is calculating the sum of the first few terms. The sum formula for an arithmetic sequence is:

• \( S_n = \frac{n}{2} (a_1 + a_n) \)

Understanding this formula is key because it lets you compute the sum of terms without adding them one by one, which is especially handy with large sequences.
In our example, we know that the sum of the first 25 terms \( S_{25} \) is 650, and the 25th term \( a_{25} \) is 62. By plugging these values into the formula, we created an equation and solved it for the first term \( a_1 \).
To simplify:

• Multiply both sides by 2 to eliminate the fraction
• Use algebra to solve for \( a_1 \)

This using of the sum formula helps connect summation and individual terms, offering a comprehensive understanding of sequences.

Nth-Term Formula

Individual terms in an arithmetic sequence are easy to find using the Nth-term formula, which is essential for understanding how sequences progress. It is given by:

• \( a_n = a_1 + (n-1) \cdot d \)

Here, \( a_n \) is the nth term, \( a_1 \) is the first term, and \( d \) is the common difference. This formula shows how each term relates to its predecessor by a constant difference.
In the current exercise, knowing \( a_1 \) allowed us to solve for the common difference \( d \) using \( a_{25} = 62 \). By substituting \( a_1 = -10 \) and \( a_{25} \) into the formula, we could unravel the pattern of the entire sequence.
This formula is pivotal, as it also helps predict further terms without listing each one.

Common Difference

The common difference, denoted as \( d \), is a vital part of arithmetic sequences. It represents how much each term increases or decreases from the previous one.
For our sequence, once we determined \( a_1 = -10 \), we used the Nth-term formula to find \(d = 3\). This tells us that each term is 3 units larger than the one before it.
Understanding the common difference allows you to recognize the pattern that the sequence follows. It's simple:

• If \( d \) is positive, the sequence grows.
• If \( d \) is negative, it decreases.
• If \( d \) is zero, all terms are the same.

This knowledge is fundamental to mastering arithmetic sequences and predicting any term in the sequence.

First Term in Sequences

The first term, \( a_1 \), is the starting point of an arithmetic sequence. It is the cornerstone of determining other terms and sums within the sequence.
In our example, calculating the first term was a crucial step. We initially used the sum formula, which led us to determine \( a_1 = -10 \). This allowed us to understand the uniqueness of the given sequence and find subsequent terms using the Nth-term formula.
This first term sets the tone for any arithmetic sequence:

• The first term plus the pattern defined by the common difference gives all the insights into the entire sequence.

Grasping the importance of \( a_1 \) is essential to unlock how sequences behave and evolve over time.