4. Examine the pattern \( 7 ; 12 ; 17 ; 22 ; \) 1. 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. l. Which term of the pattern is equal to 4177

1. The sequence is arithmetic with the first term \( a_1 = 7 \) and common difference \( d = 5 \). 2. The general formula for the \( n^{\text{th}} \) term of an arithmetic sequence is: \[ a_n = a_1 + (n-1)d \] Substituting the given values: \[ a_n = 7 + (n-1)(5) \] Simplifying, we get: \[ a_n = 7 + 5n - 5 = 5n + 2 \] 3. To determine the \( 58^{\text{th}} \) term of the pattern, substitute \( n = 58 \) into the formula: \[ a_{58} = 5(58) + 2 = 290 + 2 = 292 \] 4. To find which term of the pattern equals \( 4177 \), set up the equation: \[ 5n + 2 = 4177 \] Solving for \( n \): \[ 5n = 4177 - 2 = 4175 \] \[ n = \frac{4175}{5} = 835 \] Thus: - The general formula for the \( n^{\text{th}} \) term is \( a_n = 5n + 2 \). - The \( 58^{\text{th}} \) term is \( 292 \). - The term equal to \( 4177 \) is the \( 835^{\text{th}} \) term.