Given the arithmetk series \( 2+5+8+\cdots+89=k \) Calculate: 1. Number of terms in the series

To determine the number of terms in the arithmetic series \( 2 + 5 + 8 + \cdots + 89 \), we can use the formula for the \( n \)-th term of an arithmetic series, which is given by: \[ a_n = a_1 + (n-1)d \] Here, \( a_1 = 2 \) (the first term) and \( d = 3 \) (the common difference). We set \( a_n = 89 \) and solve for \( n \): \[ 89 = 2 + (n-1) \cdot 3 \] \[ 89 - 2 = (n-1) \cdot 3 \] \[ 87 = (n-1) \cdot 3 \] \[ n-1 = \frac{87}{3} = 29 \] \[ n = 30 \] So, the number of terms in the series is \( \boxed{30} \).