1. **Problem:** We have groups arriving with the first group having 1 person, the second group 3 people, and each subsequent group having 2 more people than the previous group. Find the number of people in the 20th 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 20th group will have \boxed{39} people. 2. **Problem:** Students learn verses each week: 1 verse after week 1, 3 after week 2, 6 after week 3, 10 after week 4. Find how many verses after 12 weeks. 3. **Observation:** The sequence 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 $n=12$:** $$T_{12} = \frac{12 \times 13}{2} = \frac{156}{2} = 78$$ 6. **Answer:** After 12 weeks, they can recite \boxed{78} verses. 3. **Problem:** A 48-inch ribbon is cut into two pieces, one piece is three times as long as the other. Find the length of each piece. 4. **Let the shorter piece be $x$. Then the longer piece is $3x$.** 5. **Equation:** $$x + 3x = 48$$ $$4x = 48$$ $$\cancel{4}x = \cancel{4}12$$ $$x = 12$$ 6. **Lengths:** Shorter piece = 12 inches Longer piece = $3 \times 12 = 36$ inches 7. **Answer:** The pieces are \boxed{12} inches and \boxed{36} inches long. 4. **Problem:** Jack is half as old as Sharon. Sharon is 3 years older than Alex. Alex and Sharon's ages add to 17. Terence is 8. Find the youngest. 5. **Let Alex's age be $A$. Then Sharon's age is $A + 3$. Jack's age is half of Sharon's: $\frac{A+3}{2}$.** 6. **Sum of Alex and Sharon:** $$A + (A + 3) = 17$$ $$2A + 3 = 17$$ $$2A = 14$$ $$\cancel{2}A = \cancel{2}7$$ $$A = 7$$ 7. **Calculate Sharon's age:** $$7 + 3 = 10$$ 8. **Calculate Jack's age:** $$\frac{10}{2} = 5$$ 9. **Terence's age is 8. Ages are:** Jack = 5, Alex = 7, Sharon = 10, Terence = 8 10. **Youngest is Jack with age \boxed{5}.** 5. **Problem:** Complete the multiplication patterns. (a) Given grid: \begin{matrix} 30 & 50 & 18 \\ 60 & 10 & 6 \\ 15 & 5 & 3 \end{matrix} (b) Given grid: \begin{matrix} 4 & 7 \\ 11 & 3 \end{matrix} (c) Given grid: \begin{matrix} 2 & 8 \\ 6 & 7 \end{matrix} (d) Given grid: \begin{matrix} 9 & 4 \\ 5 & 12 \end{matrix} **Note:** The problem does not specify what to complete or the pattern rule, so no further calculation is possible. **Summary:** - 20th group has \boxed{39} people. - After 12 weeks, \boxed{78} verses. - Ribbon pieces: \boxed{12} and \boxed{36} inches. - Youngest neighbor is Jack, age \boxed{5}.