Problem 36 - Precalculus: Mathematical for Calculus

Question

The 20 th term of an arithmetic sequence is \(101,\) and the common difference is \(3 .\) Find a formula for the \(n\) th term.

Step-by-Step Solution

Answer

The formula for the nth term is \(a_n = 3n + 41\).

Step 1: Understand the Arithmetic Sequence Formula

The general formula for the nth term of an arithmetic sequence is \( a_n = a_1 + (n-1)d \), 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: Substitute Known Values

We know that the 20th term \(a_{20} = 101\) and the common difference \(d = 3\). Substitute these into the formula: \[101 = a_1 + (20-1) \cdot 3\].

Step 3: Simplify the Equation

Simplify the expression to find an equation in terms of \(a_1\): \[101 = a_1 + 19 \times 3\] \[101 = a_1 + 57\].

Step 4: Solve for the First Term

Subtract 57 from both sides to find the first term \(a_1\): \[a_1 = 101 - 57\] \[a_1 = 44\].

Step 5: Write the nth Term Formula

Now that we know \(a_1\), we can write the formula for the nth term of the sequence: \[a_n = 44 + (n-1)\cdot 3\].

Step 6: Simplify the nth Term Formula

Distribute the common difference and simplify the formula: \[a_n = 44 + 3n - 3\] \[a_n = 3n + 41\].

Key Concepts

Arithmetic Sequence Formula

When learning about arithmetic sequences, the formula is your starting point. An arithmetic sequence consists of numbers where each term, after the first, is derived by adding a fixed number, known as the common difference, to the previous term.
The arithmetic sequence formula is given as \( a_n = a_1 + (n-1)d \). In this formula:

• \(a_n\) is the \(n\)th term you wish to find.
• \(a_1\) stands for the first term of the sequence.
• \(d\) represents the common difference between successive terms.
• \(n\) is the term position in the sequence (e.g., the 1st term, 2nd term, etc.).

Understanding this formula is crucial because it helps generate the sequence and find any term within it. With this formula, you're essentially mapping out how the sequence progresses from one term to another.

Nth Term Formula

The nth term formula is a powerful tool for identifying any specific term in an arithmetic sequence without having to write out all the preceding terms. This formula provides a direct path to the desired term, allowing for effective time management in solving problems. By substituting known values into the formula \( a_n = a_1 + (n-1)d \), you can calculate \(a_n\) directly.Consider the given exercise, where the 20th term is known to be 101. By using the nth term formula, we input:

• \(a_{20} = 101\)
• \(d = 3\)

Starting with \(a_1\) unknown, the formula helps express \(a_1\) through logical steps, eventually deriving a formula applicable to any term \(a_n\) in the sequence.

Common Difference

The common difference, represented by \(d\), is a defining characteristic of arithmetic sequences. This key value is constant throughout the sequence and is the numerical difference between any two consecutive terms.In the context of the exercise, the common difference is 3. This means each term is 3 more than the last. Recognizing and calculating the common difference assists in predicting the entire sequence's behavior and is integral to deriving the nth term formula.To find the common difference, choose any two successive terms, subtract the first from the second, and the resulting value is \(d\). The constancy of \(d\) makes arithmetic sequences straightforward to work with once you understand its role.

Solving Algebraic Equations

Solving algebraic equations is a fundamental process in mathematics, and it's especially vital when working with sequences. Once the arithmetic sequence formula is set up with known values, solving for unknown terms becomes a straightforward endeavor.In solving the exercise's equation, we start with:\[ 101 = a_1 + 19 \cdot 3 \]Simplifying this gives:\[ 101 = a_1 + 57 \]To solve for \(a_1\), subtract 57 from both sides:\[ a_1 = 101 - 57 = 44 \]Now we have all we need to finalize the nth-term formula:\[ a_n = 44 + (n-1)\cdot 3 \]Distributing and simplifying yields:\[ a_n = 3n + 41 \]Simple algebraic techniques allow you to find terms within a sequence efficiently, highlighting the power of solving equations in practical applications.