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Case Study - 1: A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track, with each round being $300$ metres. The organizers decide to increase the distance of each subsequent round by $50$ metres. The total number of rounds planned is $10$. (i) Write the fourth, fifth and sixth term of the Arithmetic Progression so formed. (ii) Determine the distance of the $8^{th}$ round. (iii) (a) Find the total distance run after completing all $10$ rounds. OR (iii) (b) If a runner completes only the first $6$ rounds, what is the total distance run by the runner?
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A.P: $300, 350, 400 \dots$ (i) $a_4 = 450, a_5 = 500, a_6 = 550$. (ii) $a_8 = 300 + 7 \times 50 = 650$ m. (iii)(a) $S_{10} = \frac{10}{2}(2 \times 300 + 9 \times 50) = 5250$ m. (iii)(b) $S_6 = \frac{6}{2}(2 \times 300 + 5 \times 50) = 2250$ m.