Problem 11

Question

$9-12$ . Find the $n$ th term of the arithmetic sequence with given first term and common difference $d$ What is the 10 the term? $$ a=\frac{5}{2}, d=-\frac{1}{2} $$

Step-by-Step Solution

Verified

Answer

The 10th term is $-2$.

1Step 1: Understand the formula for the n-th term

The n-th term of an arithmetic sequence can be found using the formula: $a_n = a + (n-1) \,d$, where $a$ is the first term, $d$ is the common difference, and $n$ is the term number.

2Step 2: Identify given values

Here, we are given the first term $a = \frac{5}{2}$ and the common difference $d = -\frac{1}{2}$. We need to find the 10th term, so $n = 10$.

3Step 3: Substitute values into the formula

Substitute the values $a = \frac{5}{2}$, $d = -\frac{1}{2}$, and $n = 10$ into the formula: \[ a_{10} = \frac{5}{2} + (10 - 1) (-\frac{1}{2}) \]

4Step 4: Simplify the expression

First, calculate the expression inside the parentheses: $10 - 1 = 9$. Then substitute back into the equation: \[ a_{10} = \frac{5}{2} + 9(-\frac{1}{2}) \].

5Step 5: Calculate the effect of the common difference

Multiply 9 by $-\frac{1}{2}$: $9 \times -\frac{1}{2} = -\frac{9}{2}$. Thus, the expression becomes: \[ a_{10} = \frac{5}{2} - \frac{9}{2} \].

6Step 6: Final calculation for the 10th term

Now, subtract $\frac{9}{2}$ from $\frac{5}{2}$: $\frac{5}{2} - \frac{9}{2} = -\frac{4}{2} = -2$. Therefore, the 10th term is $-2$.

Key Concepts

n-th term formulacommon differencefirst termsequence calculation

n-th term formula

The n-th term formula is a cornerstone in understanding arithmetic sequences. This formula helps you find any term in the sequence without needing to list all previous terms. It is defined as:

$a_n = a + (n-1)d$

Here, $a_n$ is the n-th term you want to find. $a$ represents the first term of the sequence, and $d$ is the common difference. Each time you want to find a term in the sequence, just plug the known values and solve.
Explaining it further, when you know the first term and the common difference, this formula allows you to go straight to any position $n$ in the sequence. No need to calculate each term one by one.

common difference

The common difference in an arithmetic sequence is crucial to understanding how the sequence progresses. It is the amount you add (or subtract, if negative) to go from one term to the next. In short, the common difference is what makes the sequence grow or shrink at a steady rate.

In our example, $d = -\frac{1}{2}$.

This means every term is $-\frac{1}{2}$ less than its preceding term. You can determine the common difference by subtracting any term from the term that follows it. Consistency in this difference confirms the sequence is arithmetic.
Remember, the entire sequence's behavior depends on this difference. Thus, a positive difference means the sequence increases, while a negative one results in a decreasing sequence.

first term

Understanding the first term of an arithmetic sequence is essential because it serves as the starting point for the entire sequence. The sequence builds upon this initial value.

For our example, the first term $a = \frac{5}{2}$.

This term is the foundation of the formula:

$a_n = a + (n-1)d$

Starting with this first term, every subsequent term is calculated by adding the common difference $d$ multiplied by the term position offset, $n-1$. This fundamental understanding is key to accurately applying the n-th term formula and generating any term in the sequence.

sequence calculation

Sequence calculation involves using the n-th term formula effectively to find specific terms. Let’s demonstrate with our exercise data: the first term $a = \frac{5}{2}$ and the common difference $d = -\frac{1}{2}$.