1. **Problem:** How many people will arrive in the twentieth group if the first group has 1 person, the second group has 3 people, and each subsequent group has 2 more people than the previous group? 2. **Formula:** This is an arithmetic sequence where the first term $a_1=1$ and the common difference $d=2$. 3. **General term formula:** $$a_n = a_1 + (n-1)d$$ 4. **Calculate the 20th term:** $$a_{20} = 1 + (20-1) \times 2 = 1 + 19 \times 2 = 1 + 38 = 39$$ 5. **Answer:** The twentieth group will have \boxed{39} people. 2. **Problem:** How many verses can the students recite after 12 weeks if the number of verses known at the end of weeks 1, 2, 3, and 4 are 1, 3, 6, and 10 respectively? 3. **Observation:** The sequence of verses is 1, 3, 6, 10 which are triangular numbers. 4. **Formula for the $n$th triangular number:** $$T_n = \frac{n(n+1)}{2}$$ 5. **Calculate for 12 weeks:** $$T_{12} = \frac{12 \times 13}{2} = \frac{156}{2} = 78$$ 6. **Answer:** After 12 weeks, they can recite \boxed{78} verses. 3. **Problem:** Nadia cuts a 48-inch ribbon into two pieces where one piece is three times as long as the other. Find the length of each piece. 4. **Let the shorter piece be $x$ inches. Then the longer piece is $3x$ inches.** 5. **Equation:** $$x + 3x = 48$$ 6. **Simplify:** $$4x = 48$$ $$\cancel{4}x = \cancel{4}12$$ 7. **Solve for $x$:** $$x = 12$$ 8. **Lengths:** Shorter piece = 12 inches, Longer piece = $3 \times 12 = 36$ inches. 9. **Answer:** The pieces are \boxed{12} inches and \boxed{36} inches long. 4. **Problem:** Jack is half as old as Sharon. Sharon is three years older than Alex. Alex and Sharon's ages add up to 17 years. Terence is 8 years old. Who is the youngest? 5. **Let Alex's age be $A$. Then Sharon's age is $A + 3$.** 6. **Sum of Alex and Sharon's ages:** $$A + (A + 3) = 17$$ 7. **Simplify:** $$2A + 3 = 17$$ $$2A = 14$$ $$\cancel{2}A = \cancel{2}7$$ 8. **Solve for $A$:** $$A = 7$$ 9. **Sharon's age:** $$7 + 3 = 10$$ 10. **Jack's age:** Half of Sharon's age $$= \frac{10}{2} = 5$$ 11. **Terence's age:** 8 years. 12. **Compare ages:** Jack (5), Alex (7), Terence (8), Sharon (10). 13. **Answer:** The youngest is \boxed{Jack}. 5. **Problem:** Complete the multiplication patterns. (a) Given: $$\begin{matrix} 30 & 50 & 18 \\ 60 & 10 & 6 \\ 15 & 5 & 3 \end{matrix}$$ (b) Given: $$\begin{matrix} 4 & 7 \\ 11 & 3 \end{matrix}$$ (c) Given: $$\begin{matrix} 2 & 8 \\ 6 & 7 \end{matrix}$$ (d) Given: $$\begin{matrix} 9 & 4 \\ 5 & 12 \end{matrix}$$ **Note:** The problem does not specify the pattern rule explicitly, so no further calculation is done here. **Total distinct problems:** 5