Problem 42 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{31}=5580, a_{31}=360$$

Step-by-Step Solution

Answer

\(a_1 = 0\), \(d = 12\).

Step 1: Understand the Formula for the Sum of an Arithmetic Sequence

The sum of the first \(n\) terms in an arithmetic sequence can be calculated using the formula: \[ S_n = \frac{n}{2} \times (2a_1 + (n - 1)d). \] For this problem, we have \(S_{31} = 5580\) and \(n = 31\).

Step 2: Use the Formula for the nth Term

The nth term of an arithmetic sequence is given by:\[ a_n = a_1 + (n-1)d. \]Here, \(a_{31} = 360\).

Step 3: Set Up Equations

Using the formula for the sum of the sequence, we have:\[ \frac{31}{2} \times (2a_1 + 30d) = 5580. \]And from the nth term formula, we have:\[ a_1 + 30d = 360. \]

Step 4: Solve for One Variable

From \(a_1 + 30d = 360\), solve for \(a_1\):\[ a_1 = 360 - 30d. \]

Step 5: Substitute Back into the Sum Equation

Substitute \(a_1 = 360 - 30d\) into the first equation:\[ \frac{31}{2} \times (2(360 - 30d) + 30d) = 5580. \]Simplifying, we solve:\[ 31 \times (720 - 30d) = 11160 \] \[ 22320 - 930d = 11160 \].

Step 6: Solve for \(d\)

Rearrange to solve for \(d\):\[ 22320 - 11160 = 930d \] \[ 11160 = 930d \] \[ d = \frac{11160}{930} = 12. \]

Step 7: Solve for \(a_1\)

Now that \(d = 12\), substitute back into \(a_1 = 360 - 30d\):\[ a_1 = 360 - 30(12)\] \[ a_1 = 360 - 360 = 0. \]

Key Concepts

Sum of Arithmetic Sequence

An arithmetic sequence is a list of numbers with a constant difference between consecutive terms. The sum of an arithmetic sequence up to the nth term is a frequently encountered topic in mathematics.

The formula to find the sum, denoted as \( S_n \), is:

• \( S_n = \frac{n}{2} \times (2a_1 + (n - 1)d) \)

Here, \( n \) is the number of terms, \( a_1 \) is the first term, and \( d \) is the common difference. In the given exercise, we had \( S_{31} = 5580 \) for 31 terms.

Understanding this formula is key to solving problems involving the sum of terms in an arithmetic sequence. The sum formula works by averaging the first and last terms, then multiplying by the number of terms, giving you a complete sum.

Nth Term Formula

The nth term formula helps us determine any term within an arithmetic sequence without listing all terms. The formula given is:

• \( a_n = a_1 + (n-1)d \)

Here, \( a_n \) is the nth term of the sequence, \( a_1 \) is the first term, and \( d \) is the difference between each term. In our exercise, we knew that \( a_{31} = 360 \).

By understanding and applying this formula, learners can efficiently find any position's term in a sequence. Recognizing how the term number \( n \) interacts with the first term and the common difference is essential for mastering sequences.

Solving Equations

When working with arithmetic sequences, it's common to set up and solve equations. This exercise involved solving two equations derived from sum and nth term formulas.

• The equation from the sum formula was: \( \frac{31}{2} \times (2a_1 + 30d) = 5580 \)
• The nth term equation was: \( a_1 + 30d = 360 \)

To solve these, isolate one variable and substitute back into the other equation.

In our case, we first solved for \( a_1 \) in terms of \( d, \) and then substituted into the sum equation. Solving compound equations involves algebraic skills like distribution and simplification, crucial for arithmetic sequence problems.

Arithmetic Sequence Formula

The arithmetic sequence formula is vital for understanding how sequences behave concerning their initial term and constant difference. This exercise used these formulas:

• Sum of sequence: \( S_n = \frac{n}{2} \times (2a_1 + (n - 1)d) \)
• Nth term: \( a_n = a_1 + (n-1)d \)

To find unknown values like \( a_1 \) and \( d, \) recognize their roles in these formulas. The primary task is inserting known values and manipulating equations to reveal the unknowns, as done with finding \( d = 12 \) and subsequently, \( a_1 = 0 \).

Grasping these formulas allows you to uncover patterns within a sequence, calculate specific terms, and summarize terms concisely. Using arithmetic sequence formulas is essential for students tackling numerical pattern problems.