Problem 45 - Precalculus: Mathematics for Calculus

Question

Find the indicated term of the arithmetic sequence with the given description. The 50 th term is \(1000,\) and the common difference is \(6 .\) Find the first and second terms.

Step-by-Step Solution

Answer

The first term is 706, and the second term is 712.

Step 1: Understand the Arithmetic Sequence Formula

The general formula for an arithmetic sequence is given by \( a_n = a_1 + (n-1) \cdot d \), where \( a_n \) is the nth term, \( a_1 \) is the first term, \( d \) is the common difference, and \( n \) is the term number.

Step 2: Identify Known Values

In this problem, it is given that the 50th term \( a_{50} = 1000 \) and the common difference \( d = 6 \). We need to find \( a_1 \), the first term.

Step 3: Setup the Equation for the 50th Term

Substitute the known values into the arithmetic sequence formula: \( a_{50} = a_1 + (50-1) \cdot 6 = 1000 \). Simplify to get \( a_1 + 294 = 1000 \).

Step 4: Solve for the First Term

Rearrange the equation \( a_1 + 294 = 1000 \) to solve for \( a_1 \). This gives \( a_1 = 1000 - 294 = 706 \). The first term \( a_1 \) is 706.

Step 5: Calculate the Second Term

Use the formula for the sequence to find the second term: \( a_2 = a_1 + d \). Substitute the values we've found: \( a_2 = 706 + 6 = 712 \). The second term \( a_2 \) is 712.

Key Concepts

Arithmetic Sequence Formula

An arithmetic sequence is a list of numbers where each term is obtained by adding a constant value, known as the common difference, to the previous term.
The arithmetic sequence formula is crucial for understanding how such sequences are structured and how to find any term in the sequence.
The formula is:\[a_n = a_1 + (n-1) \cdot d\]

• \( a_n \) represents the nth term of the sequence.
• \( a_1 \) is the first term.
• \( d \) is the common difference.
• \( n \) is the term number.

To find any term, you just need to know the first term \( a_1 \), the common difference \( d \), and the term number \( n \). Then plug these values into the formula, and you can easily calculate the desired term. Remember, the power of arithmetic sequences lies in their predictable pattern. That makes them very handy for solving real-life problems.

Common Difference

The common difference in an arithmetic sequence is the key to its uniform progression. It is the amount that each term increases (or decreases) by, compared to the previous term.
In mathematical terms, it is defined as the difference between any two successive terms.
Use the formula:\[d = a_2 - a_1\]The common difference \(d\) tells us how to "move" from one term to the next in the sequence.

• If \(d\) is positive, the sequence is increasing.
• If \(d\) is negative, the sequence is decreasing.
• If \(d\) is zero, all terms are the same.

In our example, with a common difference of 6, we know each subsequent term in the sequence is 6 more than the previous term. Understanding the common difference helps in predicting any part of the sequence and is foundational in forming the entire sequence given just a few terms.

Sequence Terms

In an arithmetic sequence, the terms follow a specific order determined by the initial term and the common difference. Each term is interconnected and understanding this relationship is key.- **First Term ( \( a_1 \) )**: This is the starting point of your sequence. It's the reference term for finding all others.
- **Subsequent Terms**: Each term is calculated by adding the common difference \( d \) to the previous term. For example, considering our solution where the first term \( a_1 \) is 706 and the common difference \( d \) is 6:

• The second term \( a_2 = 706 + 6 = 712 \).
• The third term would be \( 712 + 6 = 718 \), and so on.

Each term builds upon the last, and once you understand the pattern, you can quickly determine any number of terms required by simply continuing the addition process. This predictability makes arithmetic sequences straightforward and manageable.