Problem 41 - A Graphical Approach to College Algebra

Question

Find \(a_{1}\) and \(d\) for each arithmetic sequence. $$S_{20}=1090, a_{20}=102$$

Step-by-Step Solution

Answer

First term \(a_1 = 7\) and common difference \(d = 5\).

Step 1: Understanding the Arithmetic Sequence Formula

An arithmetic sequence is defined by the first term \(a_1\) and a common difference \(d\). The \(n\)-th term is given by the formula \(a_n = a_1 + (n-1)d\). The sum of the first \(n\) terms is given by \(S_n = \frac{n}{2}(a_1 + a_n)\). We are given \(S_{20} = 1090\) and \(a_{20} = 102\).

Step 2: Setting Up the Equations

From the nth-term formula, substitute \(n=20\) and \(a_{20} = 102\): \[ a_{20} = a_1 + 19d = 102 \] From the sum formula, substitute \(n=20\) and \(S_{20} = 1090\): \[ S_{20} = \frac{20}{2}(a_1 + a_{20}) = 1090 \] This simplifies to \[ 10(a_1 + 102) = 1090 \] or \[ a_1 + 102 = 109 \]

Step 3: Solving for \(a_1\)

From the equation \(a_1 + 102 = 109\), solve for \(a_1\): \[ a_1 = 109 - 102 \]\[ a_1 = 7 \]

Step 4: Solving for \(d\)

Substitute \(a_1 = 7\) into the equation from the nth-term formula: \[ a_1 + 19d = 102 \]\[ 7 + 19d = 102 \] Solve for \(d\): \[ 19d = 102 - 7 \]\[ 19d = 95 \]\[ d = \frac{95}{19} \]\[ d = 5 \]

Key Concepts

Common Difference

In an arithmetic sequence, the **common difference** is a key element that defines the progression from one term to the next. It is the constant difference between any two consecutive terms in the sequence. The common difference is denoted by the symbol **d**.

To find the common difference, subtract the first term from the second term, the second term from the third term, and so on. This value remains constant throughout the sequence.

• Example: In the sequence 3, 7, 11, ..., the common difference is calculated as 7 - 3 = 4. Hence, **d = 4**.

This consistent step forward makes the sequence orderly, and it is crucial for finding any term in the sequence.

Nth-Term Formula

The **nth-term formula** allows us to find any term in an arithmetic sequence without listing all preceding terms. The formula is expressed as:
\[ a_n = a_1 + (n-1) imes d \]
Here, **a_n** represents the nth term, **a_1** is the first term, and **d** is the common difference. **n** is the position of the term in the sequence.

• Example: For a sequence where **a_1 = 5** and **d = 3**, the 4th term, or **a_4**, would be calculated as:
  \[ a_4 = 5 + (4-1) imes 3 = 5 + 9 = 14 \]

Understanding this formula is essential for quickly determining any term in the sequence.

Sum of the First N Terms

The sum of the first **n** terms in an arithmetic sequence can be calculated using the formula:
\[ S_n = \frac{n}{2}(a_1 + a_n) \]
Here, **S_n** is the sum of the first **n** terms, **a_1** is the first term, and **a_n** is the nth term.

This formula is helpful because it lets us find the total of a large part of the sequence without having to add up each individual term. For instance, if **a_1 = 2**, **a_n = 20**, and **n = 10**, the sum is:

• \[ S_{10} = \frac{10}{2} (2 + 20) = 5 imes 22 = 110 \]

This ability to quickly sum terms is particularly useful in both academic exercises and real-world applications.

First Term of Sequence

The **first term of a sequence**, represented as **a_1**, is the initial term from which the sequence begins. It sets the starting point for the arithmetic progression. Knowing this term is vital because it acts as the foundational element of the sequence.

In exercises where the first term isn't directly given, we can often deduce it using other provided information, such as the nth term and the sum formula. For example, if we know the 20th term is 102 and the common difference is 5, we can find **a_1** using:

• \[ 102 = a_1 + (20-1) imes 5 \]
• \[ a_1 = 102 - 95 = 7 \]

Knowing the first term allows us to pick apart the sequence with both the nth-term and sum of the sequence formulas.