Problem 11 - 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?\(d .\) What is the 10 th term? $$a=\frac{5}{2}, d=-\frac{1}{2}$$

Step-by-Step Solution

Answer

The 10th term is -2.

Step 1: Identify Formula for nth term

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

Step 2: Substitute Given Values

Substitute the given values \( a = \frac{5}{2} \) and \( d = -\frac{1}{2} \) into the formula. Now, to find the 10th term, substitute \( n = 10 \) into the equation: \( a_{10} = \frac{5}{2} + (10-1) \cdot (-\frac{1}{2}) \).

Step 3: Calculate the Expression Inside Parentheses

Calculate the expression \((10-1) \cdot (-\frac{1}{2})\). This simplifies to \(9 \cdot (-\frac{1}{2}) = -\frac{9}{2}\).

Step 4: Final Calculation

Add the result from the previous step to the first term: \( a_{10} = \frac{5}{2} + (-\frac{9}{2}) = \frac{5}{2} - \frac{9}{2}\). Simplify this to get \(-2\).

Key Concepts

Understanding the nth Term Formula

The nth term formula is an essential tool when working with arithmetic sequences. It helps you calculate any term in the sequence without listing all the previous terms. The formula is:

• \( a_n = a + (n-1) \cdot d \)

Here, \( a_n \) represents the term you want to find, \( a \) is the first term, \( n \) is the term number, and \( d \) is the common difference.
To use this formula, just plug in the values for the first term, the common difference, and the term number, and perform the calculations.
This straightforward approach ensures that finding a specific term in an arithmetic sequence becomes quick and efficient.

What is Common Difference?

The common difference is a constant amount that you add to each term to get the next term in an arithmetic sequence. It's a crucial part of understanding how arithmetic sequences work.
When you know the common difference, which is denoted by \( d \), you can easily predict future terms. For example, if the common difference is negative, the sequence will decrease; if positive, it will increase.
In the given example, the common difference is \(-\frac{1}{2}\). This indicates that each term is \(\frac{1}{2}\) less than the previous one. The consistency of this difference plays a key role in the predictability and calculation of sequences.

Sequence Calculation Simplified

Sequence calculation involves determining specific terms in an arithmetic sequence using the nth term formula. Let's walk through the calculation for the 10th term.
First, substitute the known values into the formula: the first term \( a = \frac{5}{2} \), the common difference \( d = -\frac{1}{2} \), and \( n = 10 \). The formula becomes

• \( a_{10} = \frac{5}{2} + (10-1) \cdot (-\frac{1}{2}) \)

Calculate the expression inside the parentheses:

• \( 9 \cdot (-\frac{1}{2}) = -\frac{9}{2} \)

Then, perform the final arithmetic:

• \( \frac{5}{2} - \frac{9}{2} = -2 \)

This process links the theoretical understanding of sequences to practical computation, equipping you to handle similar problems with confidence.