Part A Write an equation for the $$ n $$th term of the arlthmetic sequence. What is the common difference of this sequence? $$ \begin{aligned} a_{n} & =a_{1}+(n-1) d \ & =13+(n-1)(-5) \quad a_{1}=13 ext { and } d=-5 \ & =13+(-5) n+\square\end{aligned} $$

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Part A Write an equation for the $$ n $$th term of the arlthmetic sequence. What is the common difference of this sequence? $$ \begin{aligned} a_{n} & =a_{1}+(n-1) d \ & =13+(n-1)(-5) \quad a_{1}=13 	ext { and } d=-5 \ & =13+(-5) n+\square\end{aligned} $$

UpStudy AI · Accepted Answer

Sure, let's work through **Part A** step by step.

### 1. Writing the Equation for the $$ n $$th Term

The general formula for the $$ n $$th term of an arithmetic sequence is:

$$
a_{n} = a_{1} + (n - 1)d
$$

Given:
- $$ a_{1} = 13 $$ (the first term)
- $$ d = -5 $$ (the common difference)

Substitute these values into the formula:

$$
\begin{aligned}
a_{n} &= 13 + (n - 1)(-5) \
&= 13 - 5(n - 1) \
&= 13 - 5n + 5 \quad \text{(Distribute the } -5 \text{)} \
&= (13 + 5) - 5n \
&= 18 - 5n
\end{aligned}
$$

So, the equation for the $$ n $$th term is:

$$
a_{n} = 18 - 5n
$$

Alternatively, you can express it as:

$$
a_{n} = -5n + 18
$$

### 2. Determining the Common Difference

The **common difference** ($$ d $$) is the constant amount added (or subtracted) to each term to get the next term in the sequence. From the given information:

$$
d = -5
$$

**Summary:**
- **Equation for the $$ n $$th term:** $$ a_{n} = 18 - 5n $$
- **Common difference ($$ d $$):** $$ -5 $$
The $$ n $$th term of the sequence is $$ a_{n} = 18 - 5n $$, and the common difference is $$ -5 $$.

Part A Write an equation for the \( n \)th term of the arlthmetic sequence. What is the common difference of this sequence? \( \begin{aligned} a_{n} & =a_{1}+(n-1) d \\ & =13+(n-1)(-5) \quad a_{1}=13 \text { and } d=-5 \\ & =13+(-5) n+\square\end{aligned} \)

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