Problem 21 - College Algebra

Question

Find the indicated term of the arithmetic sequence with first term, \(a_{1},\) and common difference, \(d\). Find \(a_{60}\) when \(a_{1}=35, d=-3\)

Step-by-Step Solution

Answer

The 60th term of the given arithmetic sequence (-142)

Step 1: Identify Given Values

The first term, \(a_{1}\), is 35 and the common difference, \(d\), is -3. The term to find, n, is 60. Write these down.

Step 2: Use the Arithmetic Sequence Formula

We can find any term of an arithmetic sequence using the formula: \(a_{n} = a_{1} + (n-1)*d\). Substitute the given values into this formula: \(a_{60} = 35 + (60-1)*(-3)\).

Step 3: Calculate

Working through the equation, we get: \(a_{60} = 35 + 59*(-3) = 35 - 177\).

Step 4: Finalise the Calculation

Subtracting, we find the solution: \(a_{60} = -142\). The 60th term of the sequence is -142.

Key Concepts

Common Difference

An arithmetic sequence, at its core, relies on the concept of a "common difference." This is a fundamental element that sets each term in an arithmetic sequence apart from the next.
Simply put, the common difference, denoted by \( d \), is the amount added (or subtracted) consistently from each term to reach the next. In our original exercise, the common difference is given as \( d = -3 \).
This value indicates that each term is 3 less than the one before it. For clearer understanding:

• If the common difference is positive, each subsequent term is greater than the previous one.
• If the common difference is negative, like our example, each term decreases and moves towards lower values.
• A common difference of zero would mean all terms are the same.

This regularity in change allows sequences to be predictable and enables us to formulate expressions to find specific terms.

Sequence Formula

The sequence formula for arithmetic sequences provides a mathematical rule to find any term in the sequence without listing all previous terms. This general method is expressed as \( a_{n} = a_{1} + (n-1) \cdot d \). Here’s how this formula fits into our context:

• \( a_{n} \) - Represents the specific term we aim to find.
• \( a_{1} \) - This is the first term of the sequence, which is given as 35 in our example.
• \( n \) - The term number we want to find. In our step-by-step solution, that's 60.
• \( d \) - The common difference (-3 in the example).

To find the 60th term, we substitute the known values into the formula: \( a_{60} = 35 + 59 \cdot (-3) \). It simplifies the process significantly, as we avoid manually calculating each of the previous 59 terms.

82nd Term

The concept of finding the 82nd term of an arithmetic sequence follows an identical method as we saw for the 60th term. We utilize the same sequence formula, where \( a_{1} = 35 \) and \( d = -3 \). To find the 82nd term, we set \( n = 82 \):

• Start with the sequence formula: \( a_{n} = a_{1} + (n-1) \cdot d \).
• Plug in our specific values: \( a_{82} = 35 + (82-1)(-3) \).
• Calculate: \( a_{82} = 35 + 81(-3) \).
• Simplify the equation: \( a_{82} = 35 - 243 \).
• Final result: \( a_{82} = -208 \).

Thus, the 82nd term in this arithmetic sequence is -208. This approach highlights the utility of the sequence formula, allowing us to easily forecast specific terms in the sequence without extended calculations.