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

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 \( 4170 \).