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AP Precalculus -2.1 Arithmetic and Geometric Sequences- MCQ Exam Style Questions - Effective Fall 2023

AP Precalculus -2.1 Arithmetic and Geometric Sequences- MCQ Exam Style Questions – Effective Fall 2023

AP Precalculus -2.1 Arithmetic and Geometric Sequences- MCQ Exam Style Questions – AP Precalculus- per latest AP Precalculus Syllabus.

AP Precalculus – MCQ Exam Style Questions- All Topics

Question 

 
 
 
 
 
 
 
 
 
 
 
 
Values of the terms of a geometric sequence \( g_n \) are graphed in the figure. Which of the following is an expression for the nth term of the geometric sequence?
(A) \( g_n = 4 \left( \frac{1}{2} \right)^{(n-2)} \)
(B) \( g_n = 8(2)^{(n-1)} \)
(C) \( g_n = 8 \left( \frac{1}{2} \right)^n \)
(D) \( g_n = 16 \left( \frac{1}{2} \right)^{(n-1)} \)
▶️ Answer/Explanation
Detailed solution

From the graph, likely \( g_1 = 8 \), \( g_2 = 4 \), so common ratio \( r = \frac{4}{8} = \frac{1}{2} \).
General form: \( g_n = g_1 \cdot r^{\,n-1} = 8 \cdot \left( \frac{1}{2} \right)^{n-1} \).
But none match exactly — check options:
(A) \( 4 \left( \frac{1}{2} \right)^{n-2} = 4 \cdot \left( \frac{1}{2} \right)^{n-2} = 8 \cdot \left( \frac{1}{2} \right)^{n-1} \), yes, because \( 4 \cdot \left( \frac{1}{2} \right)^{-1} = 4 \cdot 2 = 8 \).
So (A) is equivalent to the standard form.
✅ Answer: (A)

Question 

A family needs to buy one shovel and between one and eight plants, inclusive, for their new garden. The cost of the shovel is \( s \) dollars, and the cost of one plant is \( p \) dollars. The output values of which of the following give the possible costs for these items, in dollars? (Note: Assume any taxes are included in the costs.)
(A) The linear function \( C(x) = s + px \) for \( 1 \leq x \leq 8 \)
(B) The exponential function \( C(x) = s \cdot p^x \) for \( 1 \leq x \leq 8 \)
(C) The arithmetic sequence \( C_n = s + pn \) for \( 1 \leq n \leq 8 \)
(D) The geometric sequence \( C_n = s \cdot p^n \) for \( 1 \leq n \leq 8 \)
▶️ Answer/Explanation
Detailed solution

Possible costs depend on the number of plants \( n \), where \( n \) is an integer from 1 to 8.
Total cost = shovel cost + \( n \) × plant cost = \( s + pn \).
This describes a set of discrete values (not a continuous function) because \( n \) is an integer.
An arithmetic sequence with first term \( s+p \) and common difference \( p \) models this exactly: \( C_n = s + pn \), \( n = 1,2,\dots,8 \).
✅ Answer: (C)

Question 

Which of the following includes the input-output pairs \( (2,4) \) and \( (3,8) \)?
(A) The arithmetic sequence \( a_n = 4n \)
(B) The linear function \( f(n) = 2 + 4(n – 1) \)
(C) The geometric sequence \( g_n = 2^{(n-1)} \)
(D) The exponential function \( h(n) = 2 \cdot 2^{(n-1)} \)
▶️ Answer/Explanation
Detailed solution

Test each option with inputs \( n = 2 \) and \( n = 3 \):
(A) \( a_2 = 4 \cdot 2 = 8 \) ❌ (needs 4)
(B) \( f(2) = 2 + 4(2-1) = 6 \) ❌ (needs 4)
(C) \( g_2 = 2^{1} = 2 \) ❌ (needs 4)
(D) \( h(2) = 2 \cdot 2^{1} = 4 \), \( h(3) = 2 \cdot 2^{2} = 8 \) ✅
So \( h(n) = 2 \cdot 2^{(n-1)} \) matches both points.
✅ Answer: (D)

Question 

The second term of a sequence is \( 6 \), and the fourth term is \( 24 \). Of the following, which statement is true?
(A) If the sequence is geometric, the first term could be \( 1 \).
(B) If the sequence is arithmetic, the third term could be \( 12 \).
(C) If the sequence is geometric, the fifth term could be \( 48 \).
(D) If the sequence is arithmetic, the sixth term could be \( 48 \).
▶️ Answer/Explanation
Detailed solution

Let’s analyze both possibilities:
Geometric: Let \( a_2 = ar = 6 \), \( a_4 = ar^3 = 24 \). Dividing: \( r^2 = 4 \) so \( r = 2 \) or \( r = -2 \).
If \( r = 2 \), \( a = 3 \), terms: 3, 6, 12, 24, 48 → (C) says 5th term could be 48, which is true.
If \( r = -2 \), \( a = -3 \), terms: -3, 6, -12, 24, -48 → 5th term is -48, but “could be” allows for the positive case.
Arithmetic: Let \( a_2 = a + d = 6 \), \( a_4 = a + 3d = 24 \). Subtract: \( 2d = 18 \) so \( d = 9 \), \( a = -3 \). Terms: -3, 6, 15, 24, 33, 42 → (D) says 6th term could be 48, which is false.
Check (A): If geometric and first term 1, then \( r = 6 \) so 4th term \( 1 \cdot r^3 = 216 \), not 24 ❌
Check (B): Arithmetic third term: \( a_3 = a_2 + d = 6 + 9 = 15 \), not 12 ❌
✅ Answer: (C)

Question 

The general term of a sequence is given by \( a_n = 51 + 3(n – 10) \), where \( a_0 \) is the initial value. Which of the following expressions also gives the general term of the sequence?
(A) \( 10 + 3(51 – n) \)
(B) \( 17 + 3n \)
(C) \( 21 + 3n \)
(D) \( 51 \cdot 3^{(n-10)} \)
▶️ Answer/Explanation
Detailed solution

Simplify \( a_n = 51 + 3(n – 10) \):
\( a_n = 51 + 3n – 30 = 21 + 3n \).
This matches option (C).
Check initial term \( a_0 = 21 \) (since \( a_0 \) is given as the initial value in the problem statement).
✅ Answer: (C)

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