1. Problem Statement: (ii) Find an expression for the $i^{th}$ term: Given the sequence terms as $7i$. (iii) Find the 50th term. (iv) Find the sum of the original 50 terms. (vi) Find the sum of the first 50 terms of the series $70, 100, 130, \ldots$ 2. Expression for the $i^{th}$ term: The problem states the $i^{th}$ term is $7i$. So the $i^{th}$ term can be written as: $$ a_i = 7i $$ 3. The 50th term: Substitute $i = 50$ into $a_i = 7i$: $$ a_{50} = 7 \times 50 = 350 $$ 4. Sum of the original 50 terms: This is an arithmetic series with first term $a_1 = 7$ and last term $a_{50} = 350$. The sum of $n$ terms is: $$ S_n = \frac{n}{2} (a_1 + a_n) $$ Substituting values for $n = 50$: $$ S_{50} = \frac{50}{2} (7 + 350) = 25 \times 357 = 8925 $$ 5. Sum of the first 50 terms of the series $70, 100, 130, \ldots$: This sequence has first term $a_1 = 70$ and common difference $d = 100 - 70 = 30$. The $n^{th}$ term for this series is: $$ a_n = a_1 + (n-1)d = 70 + (n-1)30 $$ The 50th term: $$ a_{50} = 70 + 49 \times 30 = 70 + 1470 = 1540 $$ Sum of the first 50 terms: $$ S_{50} = \frac{50}{2} (70 + 1540) = 25 \times 1610 = 40250 $$ (Note: The remaining geometric constructions and tire survey data involve no direct calculations requested, thus they are omitted here.)