1. Problem: Find the common difference in the arithmetic sequence 5, 7, 9, 11, ... The common difference $d$ in an arithmetic sequence is found by subtracting any term from the next term: $$d = a_{n+1} - a_n$$ Calculate: $$d = 7 - 5 = 2$$ 2. Problem: Find the common ratio in the geometric sequence 3, -9, 27, -81, ... The common ratio $r$ in a geometric sequence is found by dividing any term by the previous term: $$r = \frac{a_{n+1}}{a_n}$$ Calculate: $$r = \frac{-9}{3} = -3$$ 3. Problem: Find the first four terms of an arithmetic sequence with first term $a_1 = 8$ and common difference $d = -7$ The $n$th term of an arithmetic sequence is: $$a_n = a_1 + (n-1)d$$ Calculate terms: $$a_1 = 8$$ $$a_2 = 8 + (2-1)(-7) = 8 - 7 = 1$$ $$a_3 = 8 + (3-1)(-7) = 8 - 14 = -6$$ $$a_4 = 8 + (4-1)(-7) = 8 - 21 = -13$$ 4. Problem: Find the first four terms of a geometric sequence with first term $a_1 = 24$ and common ratio $r = \frac{1}{2}$ The $n$th term of a geometric sequence is: $$a_n = a_1 \times r^{n-1}$$ Calculate terms: $$a_1 = 24$$ $$a_2 = 24 \times \left(\frac{1}{2}\right) = 12$$ $$a_3 = 24 \times \left(\frac{1}{2}\right)^2 = 24 \times \frac{1}{4} = 6$$ $$a_4 = 24 \times \left(\frac{1}{2}\right)^3 = 24 \times \frac{1}{8} = 3$$ 5. Problem: Identify the constant term in the expression $6x - 5y + 20$ The constant term is the term without any variables, which is: $$20$$ 6. Problem: Count the number of terms in the expression $18a + 6b - 2c^2 + 1$ Each distinct term separated by plus or minus is counted: Terms are $18a$, $6b$, $-2c^2$, and $1$ Number of terms = 4