37. Write a formula for the sequence below, then find $$ a_{34} $$. $$ \{-2,5,12,19, \ldots\} $$

Question

UpStudy AI · Accepted Answer

To determine a formula for the given sequence and find $$ a_{34} $$, let's analyze the sequence step by step.

**Given Sequence:**
$$ -2, 5, 12, 19, \ldots $$

### Step 1: Identify the Type of Sequence
First, observe the differences between consecutive terms:

$$
\begin{align*}
5 - (-2) &= 7 \
12 - 5 &= 7 \
19 - 12 &= 7 \
&\vdots
\end{align*}
$$

The difference between each term is constant ($$ 7 $$), indicating that this is an **arithmetic sequence**.

### Step 2: Determine the Formula for the $$ n $$-th Term
For an arithmetic sequence, the $$ n $$-th term ($$ a_n $$) can be expressed as:

$$
a_n = a_1 + (n - 1) \cdot d
$$

Where:
- $$ a_1 $$ is the first term of the sequence.
- $$ d $$ is the common difference.

**Given:**
- $$ a_1 = -2 $$
- $$ d = 7 $$

Substituting these values into the formula:

$$
\begin{align*}
a_n &= -2 + (n - 1) \cdot 7 \
&= -2 + 7n - 7 \
&= 7n - 9
\end{align*}
$$

**Formula for the $$ n $$-th term:**
$$
a_n = 7n - 9
$$

### Step 3: Find $$ a_{34} $$
Now, plug $$ n = 34 $$ into the formula to find the 34th term:

$$
\begin{align*}
a_{34} &= 7(34) - 9 \
&= 238 - 9 \
&= 229
\end{align*}
$$

**Therefore, $$ a_{34} = 229 $$.**
The formula for the sequence is $$ a_n = 7n - 9 $$, and $$ a_{34} = 229 $$.

37. Write a formula for the sequence below, then find \( a_{34} \). \[ \{-2,5,12,19, \ldots\} \]