Examine the pattern $$ 7 ; 12 ; 17 ; 22 ; $$ J. Determine a formula for the general rule for the $$ n^{ ext {th }} $$ term of the k. Determine the $$ 58^{ ext {th }} $$ term of the pattern. I. Which term of the pattern is equal to 4170

Question

Examine the pattern $$ 7 ; 12 ; 17 ; 22 ; $$ J. Determine a formula for the general rule for the $$ n^{	ext {th }} $$ term of the k. Determine the $$ 58^{	ext {th }} $$ term of the pattern. I. Which term of the pattern is equal to 4170

UpStudy AI · Accepted Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the pattern
The given sequence is $$ 7, 12, 17, 22, \ldots $$.

We can observe that:
- The first term $$ a_1 = 7 $$
- The second term $$ a_2 = 12 $$
- The third term $$ a_3 = 17 $$
- The fourth term $$ a_4 = 22 $$

To find the common difference, we can subtract consecutive terms:
- $$ 12 - 7 = 5 $$
- $$ 17 - 12 = 5 $$
- $$ 22 - 17 = 5 $$

The common difference $$ d = 5 $$.

### Step 2: Determine the formula for the $$ n^{\text{th}} $$ term
The general formula for the $$ n^{\text{th}} $$ term of an arithmetic sequence can be expressed as:
$$
a_n = a_1 + (n - 1) \cdot d
$$
Substituting the known values:
- $$ a_1 = 7 $$
- $$ d = 5 $$

The formula becomes:
$$
a_n = 7 + (n - 1) \cdot 5
$$
Simplifying this:
$$
a_n = 7 + 5n - 5 = 5n + 2
$$

### Step 3: Determine the $$ 58^{\text{th}} $$ term
Now we will calculate $$ a_{58} $$:
$$
a_{58} = 5 \cdot 58 + 2
$$

Calculating this:
$$
a_{58} = 290 + 2 = 292
$$

### Step 4: Find which term of the pattern is equal to 4170
We need to find $$ n $$ such that:
$$
a_n = 4170
$$
Using the formula we derived:
$$
5n + 2 = 4170
$$
Subtracting 2 from both sides:
$$
5n = 4168
$$
Dividing by 5:
$$
n = \frac{4168}{5} = 833.6
$$

Since $$ n $$ must be a whole number, there is no term in the sequence that equals 4170.

### Summary of Results
- The formula for the $$ n^{\text{th}} $$ term is $$ a_n = 5n + 2 $$.
- The $$ 58^{\text{th}} $$ term is $$ 292 $$.
- There is no term in the pattern that equals $$ 4170 $$.
- The $$ n^{\text{th}} $$ term formula is $$ a_n = 5n + 2 $$. - The $$ 58^{\text{th}} $$ term is 292. - There is no term equal to 4170 in the pattern.

Examine the pattern \( 7 ; 12 ; 17 ; 22 ; \) J. Determine a formula for the general rule for the \( n^{\text {th }} \) term of the k. Determine the \( 58^{\text {th }} \) term of the pattern. I. Which term of the pattern is equal to 4170

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