1. Problem: Find the missing terms in the arithmetic sequence 1, 3, 5, 7, 9, __, 13, __. 2. Explanation: This is an arithmetic sequence because the difference between consecutive terms is constant. 3. Determine common difference: Subtract a term from its successor to get $3-1=2$, so the common difference is $2$. 4. First missing term: Add $2$ to $9$ to obtain $9+2=11$. 5. Second missing term: Add $2$ to $13$ to obtain $13+2=15$. 6. Final answer: The complete sequence is 1, 3, 5, 7, 9, 11, 13, 15. 7. Problem: Find the missing terms in the geometric sequence 2, 4, 8, __, 32, __. 8. Explanation: This is geometric because each term is obtained by multiplying the previous term by the common ratio. 9. Determine ratio: Divide a term by its predecessor to get $4/2=2$, so the common ratio is $2$. 10. First missing term: Multiply $8$ by $2$ to obtain $8\times 2=16$. 11. Second missing term: Multiply $32$ by $2$ to obtain $32\times 2=64$. 12. Final answer: The complete sequence is 2, 4, 8, 16, 32, 64. 13. Problem: Explain triangular numbers and give the first five. 14. Definition: The nth triangular number is the sum of the integers from $1$ to $n$, given by the formula $$T_n=\frac{n(n+1)}{2}$$. 15. Computation: $T_1=\frac{1\cdot 2}{2}=1$. 16. Computation: $T_2=\frac{2\cdot 3}{2}=3$. 17. Computation: $T_3=\frac{3\cdot 4}{2}=6$. 18. Computation: $T_4=\frac{4\cdot 5}{2}=10$. 19. Computation: $T_5=\frac{5\cdot 6}{2}=15$. 20. Final summary: The first five triangular numbers are 1, 3, 6, 10, 15.