Problem 22 - College Algebra

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\). Find \(a_{70}\) when \(a_{1}=-32, d=4\)

Step-by-Step Solution

Answer

The 70th term of this arithmetic sequence, \(a_{70}\), is 244.

Step 1: Identify given values

We first spot that the first term of the sequence, \(a_1 = -32\), the common difference, \(d = 4\), and the term number we need to find, \(n = 70\). These values are given in the problem statement.

Step 2: Use the arithmetic sequence formula

To find the 70th term of the sequence, we apply the formula: \(a_n = a_1 + (n-1) * d\). Substituting the available values into the formula, we obtain the expression: \(a_{70} = -32 + (70-1) * 4\).

Step 3: Solve the expression

Next, we simplify the equation by performing the operations: \(a_{70} = -32 + 69 * 4\). Multiplication comes first based on the order of operations (BIDMAS/BODMAS). After performing the multiplication, we sum up the resulting value with -32.

Step 4: Get the answer

Finally, compute the above expression to arrive at the value of \(a_{70}\).

Key Concepts

First Term

In an arithmetic sequence, the first term, often denoted as \(a_1\), serves as the starting point of the sequence. It's essentially the building block from which the entire sequence is constructed. Understanding the first term is crucial because it anchors all subsequent terms through the application of the common difference. For example, if the first term is \(-32\), each additional term in the sequence begins its count from this initial value. This term is directly given in problems involving arithmetic sequences, like in our exercise where you find that \(a_1 = -32\). The first term is vital because it sets the constant foundation upon which the repetitive pattern of the sequence is based. Without it, calculating any subsequent term would be impossible.

Common Difference

The common difference, denoted as \(d\), is a key feature of an arithmetic sequence. It represents the constant amount by which each term in the sequence increases or decreases compared to the previous one. Understanding \(d\) is critical, as it dictates the sequence's uniformity and direction. When the common difference is positive, like \(d = 4\) in our exercise, it indicates the sequence is ascending. Conversely, a negative \(d\) would show a descending pattern. To visualize how \(d\) operates, consider that after the first term \(a_1 = -32\), the next term is calculated by adding the common difference: the second term would be \(a_2 = -32 + 4\). In this way, \(d\) ensures each term is evenly spaced apart, maintaining the sequence's uniformity.

Arithmetic Sequence Formula

The arithmetic sequence formula is a powerful tool that allows us to find any term in the sequence without having to compute all the previous terms. It is expressed as \(a_n = a_1 + (n-1) \times d\), where \(a_n\) represents the term you wish to find, \(a_1\) is the first term, \(n\) is the term number, and \(d\) is the common difference. This formula is derived by noticing that each term increments by the common difference \(d\) a specific number of times to reach the desired position. For instance, if we want to find the 70th term, \(a_{70}\), given \(a_1 = -32\) and \(d = 4\), the formula becomes \(-32 + (70-1) \times 4\). Steps in using the formula include:

• Substitute all known values into the formula.
• Simplify within the parentheses as needed.
• Perform the multiplication and addition to find the term.

Through this structured approach, calculating any term in the sequence becomes straightforward and efficient.