1. **Problem 203:** Find the sum of all even integers between 99 and 301. The formula for the sum of the first $n$ positive integers is given by: $$ S_n = \frac{n(n+1)}{2} $$ We want the sum of even integers between 99 and 301, i.e., from 100 to 300. 2. **Identify the even integers:** The even integers from 100 to 300 are: 100, 102, 104, ..., 300. 3. **Count the number of terms:** Number of even integers = $\frac{300 - 100}{2} + 1 = \frac{200}{2} + 1 = 100 + 1 = 101$ terms. 4. **Sum of an arithmetic sequence:** Sum = $\frac{\text{number of terms} \times (\text{first term} + \text{last term})}{2}$ Sum = $\frac{101 \times (100 + 300)}{2} = \frac{101 \times 400}{2} = 101 \times 200 = 20,200$ --- 5. **Problem 204:** Find the day of the week on November 16, 2014, given November 16, 2001 was a Friday. 6. **Calculate total days between Nov 16, 2001 and Nov 16, 2014:** Years: 2002 to 2014 inclusive = 13 years. Leap years with 366 days: 2004, 2008, 2012 (3 leap years) Non-leap years with 365 days: 13 - 3 = 10 years Total days = $3 \times 366 + 10 \times 365 = 1098 + 3650 = 4748$ 7. **Calculate day of the week:** Number of weeks = $\lfloor \frac{4748}{7} \rfloor = 678$ weeks and remainder days = $4748 - 678 \times 7 = 4748 - 4746 = 2$ days. Starting from Friday, adding 2 days: Friday + 1 day = Saturday Saturday + 1 day = Sunday So, November 16, 2014 was a **Sunday**. --- 8. **Problem 205:** How many prime numbers between 1 and 100 are factors of 7,150? 9. **Prime factorization of 7,150:** Start dividing by smallest primes: $7,150 \div 2 = 3,575$ $3,575 \div 5 = 715$ $715 \div 5 = 143$ $143 \div 11 = 13$ $13 \div 13 = 1$ Prime factors: 2, 5, 11, 13 All are prime numbers between 1 and 100. Number of such primes = 4 --- 10. **Problem 206:** Given sequence $a_1=3$, $a_2=5$, and for $n>2$, $a_n = a_1 a_2 \cdots a_{n-1}$. Given $a_n = t$ for some $n > 2$, find $a_{n+2}$ in terms of $t$. 11. **Analyze the sequence:** $a_3 = a_1 a_2 = 3 \times 5 = 15$ $a_4 = a_1 a_2 a_3 = 3 \times 5 \times 15 = 225$ $a_5 = a_1 a_2 a_3 a_4 = 3 \times 5 \times 15 \times 225 = 33,750$ 12. **Express $a_n$ in terms of $a_{n-1}$:** Since $a_n = \prod_{k=1}^{n-1} a_k$, then $a_{n+1} = \prod_{k=1}^n a_k = a_n \times a_n = (a_n)^2 = t^2$ Similarly, $a_{n+2} = \prod_{k=1}^{n+1} a_k = a_{n+1} \times a_n = t^2 \times t = t^3$ **Final answers:** 203: 20,200 (Option B) 204: Sunday (Option A) 205: Four (Option D) 206: $a_{n+2} = t^3$ (Option C)