A state government installs smart street lights along a newly constructed expressway. The first light pole is placed 45 \(m\) from the entrance gate. Each subsequent pole is installed 12 \(m\) farther than the previous one, maintaining a uniform spacing throughout the highway. The project continues until 40 poles have been installed. The engineers maintain a record of the distances (in metres) of the poles from the entrance gate.
(i) Represent the distances of the poles from the entrance gate as an arithmetic progression.
(ii) Obtainthe distance of last pole from the entrance gate.
(iii) A maintenance engineer is assigned to inspect the middle pole(s) along the expressway. Compute the pole number(s) to be inspected.
(iv) During an inspection, another engineer is instructed to inspect the \(27^{\text{th}}\) pole. Determine its distance from the entrance gate.