1. **Problem statement:** We have a theatre seating arrangement where row 1 has 28 seats, and each subsequent row has one more seat than the previous row. 2. **Find the number of seats in row 10:** The number of seats in row $n$ is given by the arithmetic sequence formula: $$T_n = a + (n-1)d$$ where $a=28$ (seats in row 1) and $d=1$ (increment per row). Calculate $T_{10}$: $$T_{10} = 28 + (10-1) \times 1 = 28 + 9 = 37$$ So, row 10 has **37 seats**. 3. **Find the number of rows if the last row has 50 seats:** Set $T_n = 50$ and solve for $n$: $$50 = 28 + (n-1) \times 1$$ $$50 - 28 = n - 1$$ $$22 = n - 1$$ $$n = 23$$ So, there are **23 rows** in the theatre. 4. **Find the total number of seats in the theatre:** The total number of seats is the sum of the arithmetic series: $$S_n = \frac{n}{2} [2a + (n-1)d]$$ Substitute $n=23$, $a=28$, $d=1$: $$S_{23} = \frac{23}{2} [2 \times 28 + (23-1) \times 1]$$ $$= \frac{23}{2} [56 + 22] = \frac{23}{2} \times 78$$ $$= 23 \times 39 = 897$$ So, the theatre has **897 seats in total**. 5. **Find $n$ and the number of people in the next row when 600 people attend:** We want to find the number of full rows $n$ such that the total seats in $n$ rows is less than or equal to 600, and the next row has some people seated. Sum of $n$ rows: $$S_n = \frac{n}{2} [2a + (n-1)d] = \frac{n}{2} [56 + (n-1)] = \frac{n}{2} (55 + n)$$ We want $S_n \leq 600$ and $S_{n+1} > 600$. Test $n=10$: $$S_{10} = \frac{10}{2} (55 + 10) = 5 \times 65 = 325$$ Test $n=15$: $$S_{15} = \frac{15}{2} (55 + 15) = 7.5 \times 70 = 525$$ Test $n=16$: $$S_{16} = \frac{16}{2} (55 + 16) = 8 \times 71 = 568$$ Test $n=17$: $$S_{17} = \frac{17}{2} (55 + 17) = 8.5 \times 72 = 612$$ Since $S_{16} = 568 \leq 600$ and $S_{17} = 612 > 600$, the first 16 rows are fully occupied. Number of people in the next row (row 17): $$600 - 568 = 32$$ Row 17 has $T_{17} = 28 + (17-1) = 28 + 16 = 44$ seats. So, **$n=16$ full rows** and **32 people seated in row 17**. **Final answers:** - (a) 37 seats in row 10 - (b) 23 rows in total - (c) 897 total seats - (d) $n=16$ full rows, 32 people in next row