Question 1: Write the formula for the n-th term of a geometric sequence and define each component.

Answer: a_n = a_1 × r^{n-1} where a_n is the n-th term, a_1 is the first term, r is the common ratio, and n is the term number.

Question 2: Calculate the 5th term of a geometric sequence where a_1=3 and r=2.

Answer: 48

Question 3: A student incorrectly applies the formula and calculates the 4th term as 24 when a_1=3 and r=2. Identify the mistake.

Answer: They used r^3 instead of r^{n-1}, calculating r^3=8 instead of r^{4-1}= r^3=8, but perhaps they forgot to multiply by a_1 or misapplied the formula.

Question 4: Construct the expression for the 7th term of a geometric sequence with a_1=5 and common ratio r=3.

Answer: a_7=5×3^{6}

Question 5: Plot the graph of y=2×(3^{n-1}) for n=1 to 6 on the grid.

Answer: Students will plot points (n, y) for n=1 to 6, with y=2, 6, 18, 54, 162, 486

Question 6: A sequence has first term a_1=4 and ratio r=0.5. Find the 6th term.

Answer: 0.125

Question 7: Explain why the formula a_n = a_1 × r^{n-1} can lead to misconceptions if students forget to include the exponent.

Answer: Omitting the exponent can lead students to believe the ratio is multiplied directly without accounting for the position in the sequence, resulting in incorrect terms.

Question 8: Identify and correct the mistake: Given a_1=2, r=3, the student computes the 4th term as 81.

Answer: They used r^4 instead of r^{n-1}; correct calculation: a_4=2×3^{3}=54.

Question 9: Derive the formula for the sum of the first n terms of a geometric sequence.

Answer: S_n = a_1(1−r^n)/(1−r), r≠1

Question 10: A sequence is geometric with a_1=5 and r=−2. Find the 8th term and explain any sign pattern.

Answer: a_8=5×(−2)^{7} = 5×(−128) = -640

Question 11: Common mistake: A student calculates the 3rd term as 12 when a_1=4 and r=3. What did they do wrong?

Answer: They likely used r^2 instead of r^{n-1} (which should be r^2), and forgot to multiply by a_1 or misapplied the exponents.

Question 12: Challenge: Given a geometric sequence with a_1=1 and r=0.5, find n such that a_n<0.01.

Answer: Solve 1×0.5^{n−1}<0.01 → 0.5^{n−1}<0.01 → n−1>log_{0.5}(0.01) ≈6.64 → n≥8.