(f) \( 1 \frac{1}{2} ; 2 ; 2 \frac{1}{2} ; \ldots \) is a given sequence. (1) Determine the general rule ( \( n^{\text {th }} \) term). (2) Calculate the \( 70^{\text {th }} \) term. (3) Calculate the \( 85^{\text {th }} \) term. (4) Which term equals 151 ? (5) Which term is equal to \( 45 \frac{1}{2} \) ?

1. The given sequence is arithmetic with first term \[ a_1 = 1 \frac{1}{2} = \frac{3}{2} \] and common difference \[ d = \left(2 - \frac{3}{2}\right) = \frac{1}{2}. \] Thus, the \( n^{\text{th}} \) term is \[ a_n = a_1 + (n-1)d = \frac{3}{2} + (n-1)\left(\frac{1}{2}\right). \] Simplify: \[ a_n = \frac{3}{2} + \frac{n}{2} - \frac{1}{2} = \frac{n + 2}{2}. \] 2. To calculate the \( 70^{\text{th}} \) term, substitute \( n = 70 \) into the formula: \[ a_{70} = \frac{70 + 2}{2} = \frac{72}{2} = 36. \] 3. For the \( 85^{\text{th}} \) term, substitute \( n = 85 \): \[ a_{85} = \frac{85 + 2}{2} = \frac{87}{2} = 43 \frac{1}{2}. \] 4. To determine which term equals 151, set \( a_n = 151 \): \[ \frac{n + 2}{2} = 151. \] Multiply both sides by 2: \[ n + 2 = 302. \] Subtract 2 from both sides: \[ n = 300. \] Thus, the \( 300^{\text{th}} \) term is 151. 5. To find the term equal to \( 45 \frac{1}{2} \), set \( a_n = 45 \frac{1}{2} = \frac{91}{2} \): \[ \frac{n + 2}{2} = \frac{91}{2}. \] Multiply both sides by 2: \[ n + 2 = 91. \] Subtract 2 from both sides: \[ n = 89. \] Thus, the term \( 45 \frac{1}{2} \) is the \( 89^{\text{th}} \) term.