Problem 13 - Precalculus: Mathematics for Calculus

Question

Find the \(n\) th term of the arithmetic sequence with given first term \(a\) and common difference \(d .\) What is the 10 th term? $$a=-0.7, \quad d=-0.2$$

Step-by-Step Solution

Answer

The 10th term of the sequence is -2.5.

Step 1: Understand the Formula for the Arithmetic Sequence

The formula for the nth term of an arithmetic sequence is given by \( a_n = a + (n-1) \times d \), where \( a \) is the first term, \( d \) is the common difference, and \( n \) is the term number you want to find.

Step 2: Identify Given Values

From the problem, we know the first term \( a = -0.7 \) and the common difference \( d = -0.2 \). We need to find the 10th term, so \( n = 10 \).

Step 3: Substitute Values into Formula

Substitute \( a = -0.7 \), \( d = -0.2 \), and \( n = 10 \) into the formula: \( a_{10} = -0.7 + (10-1) \times (-0.2) \).

Step 4: Simplify the Calculation

First, calculate the product of the common difference and \( (n-1) \): \((10-1) \times (-0.2) = 9 \times (-0.2) = -1.8\).

Step 5: Compute the 10th Term

Add the result from step 4 to the first term: \(-0.7 + (-1.8) = -2.5\). Therefore, the 10th term is \(-2.5\).

Key Concepts

nth term formula

An arithmetic sequence is a sequence of numbers where each term after the first is derived by adding a fixed, constant value, called the common difference. To determine any term's position in this sequence, we use the nth term formula. The formula is very straightforward: \( a_n = a + (n-1) \times d \). Here, \( a_n \) represents the nth term, \( a \) is the first term, \( d \) stands for the common difference, and \( n \) is the term position in the sequence.
This formula is highly useful because it allows you to calculate any term in the sequence without having to write out all the preceding terms. Whether you're asked for the 10th term or the 1000th term, this formula comes to the rescue.
Imagine you're reading a never-ending story but you want to skip to the climax. In arithmetic sequences, the nth term formula lets you quickly find that climactic spot without flipping through every page. Knowing this formula is like having a shortcut through the tale of numbers!

common difference

In arithmetic sequences, the common difference is the magic ingredient that keeps the sequence steady and predictable. It's the amount you consistently add (or subtract) to move from one term to the next. In mathematical terms, the common difference \( d \) is found by subtracting the first term from the second or any subsequent term from the term that follows it.

Calculating the Common Difference

To calculate it, take any term in the sequence and subtract it from the next one.

• If \( a_1 \) is the first term and \( a_2 \) is the second term, then \( d = a_2 - a_1 \).
• Ensures that the pattern of the sequence is uniform.

In the problem at hand, the common difference is \( -0.2 \). This negative value indicates that our sequence decreases by 0.2 with each subsequent term. It's like walking down steps, where each step down is exactly 0.2 lower than the last.

term calculation

Calculating a term in an arithmetic sequence requires just a pinch of arithmetic. To find a specific term, you simply substitute the known values into the nth term formula and solve the equation.

Step-by-Step Calculation

Using the known values:

• First term \( a = -0.7 \)
• Common difference \( d = -0.2 \)
• Sequence position \( n = 10 \)

Substitute them into the formula: \( a_n = a + (n-1) \times d \). For the 10th term, replace \( a \), \( d \), and \( n \) to get \( a_{10} = -0.7 + (10-1) \times (-0.2) \).

Breaking Down the Calculation

• First, calculate \((10-1) \times (-0.2) = -1.8\).
• Then, add this product to the first term: \( -0.7 + (-1.8) = -2.5 \).

This method reduces the potential for errors and results in the 10th term of \( -2.5 \). Each calculation step sticks to the basics of arithmetic, ensuring you get the right term without hassle. "

Verifying Term Consistency

Conduct a quick check by calculating previous terms in the sequence using the same formula to see if they align with the expected results. This consistency confirms the formula's accuracy.