Problem 35 - Contemporary Precalculus

Question

Use the given information about the arithmetic sequence with common difference d to find a and a formula for \(a_{n}\). $$a_{3}=3, d=5$$

Step-by-Step Solution

Answer

Answer: The first term of the arithmetic sequence is -7, and the formula for the nth term of the sequence is \(a_n = 5n - 12\).

Step 1: Write down the formula for the nth term of an arithmetic sequence

The formula for the nth term of an arithmetic sequence is as follows: $$a_n = a + (n-1)d$$

Step 2: Plug in the given information to find a

The given information is \(a_3 = 3\) and \(d = 5\). We can plug these values into the formula from Step 1: $$3 = a + (3-1)(5)$$

Step 3: Solve for a

Now we can solve the equation from Step 2 for \(a\): $$3 = a + (2)(5)$$ $$3 = a + 10$$ $$a = -7$$

Step 4: Write down the formula for \(a_{n}\) with the values of a and d

Now that we have the value for \(a\), we can plug it into the formula for the nth term to create a formula for the arithmetic sequence: $$a_n = -7 + (n-1)(5)$$

Step 5: Simplify the \(a_n\) formula

Let's simplify the \(a_n\) formula by distributing the 5: $$a_n = -7 + 5(n-1)$$ $$a_n = -7 + 5n - 5$$ $$a_n = 5n - 12$$ That's it! The first term of the arithmetic sequence is \(a = -7\), and the formula for the nth term of the sequence is \(a_n = 5n - 12\).

Key Concepts

Nth Term Formula

Understanding how to find any term in an arithmetic sequence is crucial. This is where the nth term formula comes in handy. The formula for an arithmetic sequence is given by \( a_n = a + (n-1)d \). Here, \( a_n \) represents the nth term of the sequence, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference. This formula essentially tells you how many steps (or common differences) to move from the starting point (the first term) to reach the term you're interested in.

• The formula helps you quickly find any term without listing all the previous ones.
• It is especially useful for solving mathematical problems involving sequences.

By applying this formula, you can efficiently handle arithmetic sequences in exams and homework tasks.

Common Difference

In any arithmetic sequence, the common difference \( d \) plays a pivotal role. It is the amount added (or subtracted) consistently to obtain the next term of the sequence. Recognizing this difference allows us to understand the pattern of the sequence.

• The common difference can be calculated by subtracting the first term from the second term of the sequence.
• If the common difference is positive, the sequence is increasing.
• If the common difference is negative, the sequence is decreasing.

A clear understanding of how to determine and use the common difference is essential for a deeper grasp of arithmetic sequences. When given \( d = 5 \), it means each subsequent term in the sequence increases by 5.

Sequence Formula

The sequence formula is a backbone for understanding the sequence's construction. In context, the sequence formula for an arithmetic sequence is derived from the nth term formula. For instance, starting with \( a = -7 \) and \( d = 5 \), the sequence formula becomes \( a_n = -7 + (n-1)(5) \).

• It offers a direct way to calculate specific terms, like \( a_3 \) (the third term).
• The formula simplifies to \( a_n = 5n - 12 \), providing a clear linear relationship between the term number \( n \) and the value of the term \( a_n \).

Once you have the sequence formula, you're equipped to tackle any term number you need. This understanding demystifies how arithmetic sequences are structured.

Mathematical Problem Solving

Problem-solving in mathematics often includes working with sequences and series. Arithmetic sequences provide a prime example of this, as they involve logical steps and formula manipulation, crucial skills in math.

• The process: identify known values, apply the nth term formula, and solve for unknowns.
• Once you understand how to manipulate these elements, you can approach more complex problems with confidence.
• This enhances your analytical skills, making math not just about numbers, but understanding patterns and structures.

By breaking down problems and utilizing foundational concepts like arithmetic sequences, mathematical problem solving becomes an engaging and manageable challenge.