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)

Question

In a certain simulation, the population of a bacteria colony can be modeled using a geometric sequence, where the first day of the simulation is day 1. The population on day 4 was 4,000 bacteria, and the population on day 8 was 49,000 bacteria. What was the population of the colony on day 6 based on the simulation?

(A) 26,500
(B) 26,192
(C) 14,000
(D) 611

▶️ Answer/Explanation

Detailed solution

Geometric sequence: $ a_n = a_1 r^{n-1} $.
Given $ a_4 = 4000 $, $ a_8 = 49000 $.
\[ a_8 = a_4 r^{4} \quad \Rightarrow \quad 49000 = 4000 r^4 \]
\[ r^4 = \frac{49000}{4000} = 12.25 \]
\[ r = (12.25)^{1/4} = \sqrt{\sqrt{12.25}} = \sqrt{3.5} \approx 1.870828693 \]
Then $ a_6 = a_4 r^{2} = 4000 \times (12.25)^{1/2} = 4000 \times 3.5 = 14000 $.
✅ Answer: (C)

Question

A geometric sequence has $a_5 = 64$ and $a_4 = 32$. Write an explicit formula to represent this sequence.

a. $a_n = 3 \cdot \left(\frac{1}{2}\right)^{n-1}$

b. $a_n = 3 \cdot 2^{n-1}$

c. $a_n = 4 \cdot 2^{n-1}$

d. $a_n = 3 \cdot \left(\frac{3}{5}\right)^{n-1}$

▶️ Answer/Explanation

Detailed solution

The common ratio is $r = \frac{a_5}{a_4} = \frac{64}{32} = 2$.
Using the formula $a_n = a_1 \cdot r^{n-1}$, we find $a_4 = a_1 \cdot 2^{4-1}$.
Substitute the known values: $32 = a_1 \cdot 2^3$, which means $32 = a_1 \cdot 8$.
Solving for the first term gives $a_1 = \frac{32}{8} = 4$.
The explicit formula is $a_n = 4 \cdot 2^{n-1}$.
Comparing this to the given options, the correct choice is c.

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 $ n $th term of the geometric sequence?

(A) $ \frac{1}{4}\left(\frac{1}{2}\right)^n $

(B) $ \frac{1}{2} \cdot 2^n $

(D) $ 4(2)^{n-4} $

▶️ Answer/Explanation

Detailed solution

From the graph, we observe the points: $ (2, 1) $, $ (3, 2) $, $ (4, 4) $, and $ (5, 8) $.
This shows a geometric sequence with a common ratio of $ 2 $ (e.g., $ \frac{4}{2} = 2 $).
We test the options by substituting $ n = 2 $, knowing that $ g_2 $ must equal $ 1 $:
(A) $ \frac{1}{4}\left(\frac{1}{2}\right)^2 = \frac{1}{4} \cdot \frac{1}{4} = \frac{1}{16} $ (Incorrect)
(B) $ \frac{1}{2} \cdot 2^2 = \frac{1}{2} \cdot 4 = 2 $ (Incorrect)
(C) $ (2)^{2-1} = 2^1 = 2 $ (Incorrect)
(D) $ 4(2)^{2-4} = 4(2)^{-2} = 4\left(\frac{1}{4}\right) = 1 $ (Correct)
Therefore, the correct expression is $ 4(2)^{n-4} $.

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 $n$th term of the geometric sequence?

(A) $9\left(\frac{2}{3}\right)^n$

(B) $6\left(\frac{2}{3}\right)^{n-1}$

(D) $4\left(\frac{2}{3}\right)^{n-3}$

▶️ Answer/Explanation

Detailed solution

From the graph, we can identify the coordinates of the first few terms: $(1, 9)$, $(2, 6)$, and $(3, 4)$.
This gives us the first term $a_1 = 9$ and the second term $a_2 = 6$.
Calculate the common ratio $r$ of the geometric sequence: $r = \frac{a_2}{a_1} = \frac{6}{9} = \frac{2}{3}$.
The standard formula for the $n$th term is $g_n = a_1 \cdot r^{n-1}$, which gives $g_n = 9\left(\frac{2}{3}\right)^{n-1}$.
To find the matching option, we can substitute a known point like $n=3$ (where $g_3 = 4$) into the choices.
Checking Option (D): $4\left(\frac{2}{3}\right)^{3-3} = 4\left(\frac{2}{3}\right)^0 = 4(1) = 4$. This matches the graph perfectly.
Algebraically, we can also rewrite the standard formula to match Option (D): $9\left(\frac{2}{3}\right)^{n-1} = 9\left(\frac{2}{3}\right)^2 \left(\frac{2}{3}\right)^{n-3} = 9 \cdot \frac{4}{9} \cdot \left(\frac{2}{3}\right)^{n-3} = 4\left(\frac{2}{3}\right)^{n-3}$.
Thus, the correct expression is (D).

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 $n$th term of the geometric sequence?

(A) $-4{\left(\frac{1}{2}\right)}^{n-1}$

(B) $4{\left(-\frac{1}{2}\right)}^{n-1}$

(D) $8{\left(-\frac{1}{2}\right)}^{n}$

▶️ Answer/Explanation

Detailed solution

From the graph, we identify the coordinates of the first few terms as $(1, 4)$, $(2, -2)$, and $(3, 1)$.
The first term of the sequence is $g_1 = 4$.
We find the common ratio $r$ by dividing the second term by the first: $r = \frac{g_2}{g_1} = \frac{-2}{4} = -\frac{1}{2}$.
Verifying with the third term: $g_3 = -2 \times (-\frac{1}{2}) = 1$, which matches the graph.
The general formula for a geometric sequence is $g_n = a_1 \cdot r^{n-1}$.
Substituting our values, we get $g_n = 4 \cdot {\left(-\frac{1}{2}\right)}^{n-1}$.
Comparing this result with the options, it matches option (B).

Correct Answer: (B)

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 $ n $th term of the geometric sequence?

(A) $ 5\left(\frac{1}{2}\right)^{n/2} $

(B) $ 5\left(\frac{1}{2}\right)^{n-2} $

(D) $ 10\left(\frac{1}{2}\right)^{2n} $

▶️ Answer/Explanation

Detailed solution

From the graph, we identify two key points: $ (0, 10) $ and $ (2, 5) $. This means $ g_0 = 10 $ and $ g_2 = 5 $.

We test each option by substituting $ n = 0 $ first to check the initial value:

(A) $ n=0 \to 5(1/2)^0 = 5 $. Incorrect, as $ g_0 \neq 10 $.

(B) $ n=0 \to 5(1/2)^{-2} = 5(4) = 20 $. Incorrect, as $ g_0 \neq 10 $.

(C) $ n=0 \to 10(1/2)^0 = 10 $. Correct. Now check $ n=2 \to 10(1/2)^1 = 5 $. Correct.

(D) $ n=0 \to 10(1/2)^0 = 10 $. Correct. Now check $ n=2 \to 10(1/2)^4 = 10/16 \neq 5 $. Incorrect.

Thus, the correct expression matches the graph values in option (C).

Question

A large theater has rows of seats arranged in a way that the number of seats in each consecutive row form an arithmetic sequence. If the fourth row contains $30$ seats and the eighth row contains $54$ seats, which of the following gives the number of seats in the tenth row?

(A) $60$

(B) $66$

(D) $78$

▶️ Answer/Explanation

Detailed solution

Let the number of seats in row $n$ be denoted by $a_n$ and the common difference by $d$.

We are given $a_4 = 30$ and $a_8 = 54$.

Using the arithmetic sequence formula differences $a_8 – a_4 = (8-4)d$:

$54 – 30 = 4d \Rightarrow 24 = 4d \Rightarrow d = 6$.

To find the number of seats in the tenth row ($a_{10}$), we calculate:

$a_{10} = a_4 + (10-4)d$

$a_{10} = 30 + 6(6) = 30 + 36 = 66$.

Therefore, the correct answer is (B).

Question

Let $ a_n $ represent an arithmetic sequence where $ a_3 = 22 $ and $ a_6 = 10 $. What is the value of $ a_{12} $?

(A) $ -48 $

(B) $ -14 $

(D) $ -2 $

▶️ Answer/Explanation

Detailed solution

The relationship between any two terms in an arithmetic sequence is $ a_m = a_n + (m-n)d $.

First, substitute the given values $ a_6 = 10 $ and $ a_3 = 22 $ to find the common difference $ d $:

$ 10 = 22 + (6 – 3)d \implies 10 = 22 + 3d $.

Solving for $ d $: $ 3d = 10 – 22 = -12 \implies d = -4 $.

Now, solve for $ a_{12} $ using the term $ a_3 $ and the difference $ d = -4 $:

$ a_{12} = a_3 + (12 – 3)d = 22 + 9(-4) $.

$ a_{12} = 22 – 36 = -14 $.

Therefore, the correct option is (B).

Question

$n$	0	1	2	3	4
$s_n$	12	6	0	$-6$	$-12$

The table gives values of the sequence $s_n$ at selected values of $n$. Which of the following statements about $s_n$ is true?

(A) $s_n$ could be an arithmetic sequence, because successive terms have a constant difference.

(B) $s_n$ could be an arithmetic sequence, because successive terms have constant proportional change.

(D) $s_n$ could be a geometric sequence, because successive terms have constant proportional change.

▶️ Answer/Explanation

Detailed solution

To identify the type of sequence, we analyze the relationship between consecutive terms in the table.
First, calculate the difference between the first two terms: $6 – 12 = -6$.
Next, calculate the difference between the second and third terms: $0 – 6 = -6$.
Checking the remaining terms confirms the pattern: $-6 – 0 = -6$ and $-12 – (-6) = -6$.
Since the difference between successive terms is a constant value ($d = -6$), the sequence is arithmetic.
By definition, an arithmetic sequence is characterized by a constant difference, not proportional change.
Therefore, the correct statement is that $s_n$ could be an arithmetic sequence due to the constant difference.

Question

$n$	2	3	4	5	6
$s_n$	1	2	4	8	16

The table gives values of the sequence $s_n$ at selected values of $n$. Which of the following statements about $s_n$ is true?

(A) $s_n$ could be an arithmetic sequence, because successive terms have a constant difference.

(B) $s_n$ could be an arithmetic sequence, because successive terms have constant proportional change.

(D) $s_n$ could be a geometric sequence, because successive terms have constant proportional change.

▶️ Answer/Explanation

Detailed solution

To determine the type of sequence, we analyze the relationship between consecutive terms of $s_n$: $1, 2, 4, 8, 16$.

First, check for an arithmetic sequence by looking for a constant difference: $2-1=1$, $4-2=2$. Since $1 \neq 2$, it is not arithmetic.

Next, check for a geometric sequence by looking for a constant ratio (proportional change): $\frac{2}{1} = 2$, $\frac{4}{2} = 2$, $\frac{8}{4} = 2$.

The sequence has a constant common ratio $r = 2$.

A sequence with a constant proportional change between terms is defined as a geometric sequence.

Therefore, $s_n$ is a geometric sequence because successive terms have a constant proportional change.

Correct Option: (D)

Question

$n$	$1$	$5$	$7$	$8$	$15$
$a_n$	$b$	$32$	$26$	$c$	$2$

The table above contains selected values of an arithmetic sequence $a_n$, where $b$ and $c$ are constants. What is the value of $b + c$?

(A) $49$

(B) $58$

(D) $76$

▶️ Answer/Explanation

Detailed solution

First, find the common difference $d$ using the known terms $a_5 = 32$ and $a_7 = 26$:
$a_7 = a_5 + 2d \implies 26 = 32 + 2d \implies -6 = 2d \implies d = -3$.

Next, calculate $b$ (which corresponds to $a_1$) by working backwards from $a_5$:
$b = a_1 = a_5 – 4d = 32 – 4(-3) = 32 + 12 = 44$.

Then, calculate $c$ (which corresponds to $a_8$) using $a_7$ and $d$:
$c = a_8 = a_7 + d = 26 + (-3) = 23$.

Finally, calculate the value of $b + c$:
$b + c = 44 + 23 = 67$.

Question

Let $ g_n $ be a geometric sequence with $ g_1 = 3 $ and $ g_4 = 24 $. Which of the following is the value of $ g_3 $?

(A) $ 6 $

(B) $ 8 $

(D) $ 17 $

▶️ Answer/Explanation

Detailed solution

The general formula for the $ n $-th term of a geometric sequence is $ g_n = g_1 \cdot r^{n-1} $.
We are given $ g_1 = 3 $ and $ g_4 = 24 $. Substituting these into the formula:
$ 24 = 3 \cdot r^{4-1} \implies 24 = 3r^3 $.
Dividing both sides by 3 gives $ r^3 = 8 $.
Solving for the common ratio, we get $ r = 2 $.
Now, we calculate $ g_3 $ using $ r = 2 $ and $ n = 3 $:
$ g_3 = 3 \cdot 2^{3-1} = 3 \cdot 2^2 $.
$ g_3 = 3 \cdot 4 = 12 $.

Question

Given the sequence whose second term is $ ae^2 $ and $ ae^5 $, which of the following equations can represent the $ k $th term of this sequence?

(A) $ g_k = a(e^k) $

(B) $ g_k = a(e)^{k-1} $

(D) $ g_k = a(e+k)^{k-1} $

▶️ Answer/Explanation

Detailed solution

We need to find the equation $ g_k $ such that the second term (where $ k=2 $) equals $ ae^2 $. We verify this by substituting $ k=2 $ into each option.

(A) $ g_2 = a(e^2) = ae^2 $. This matches the given term exactly. (Also, $ k=5 $ yields $ ae^5 $).

(B) $ g_2 = a(e)^{2-1} = ae^1 = ae $. This is incorrect.

(C) $ g_2 = a(e^2) – 2 $. This is incorrect.

(D) $ g_2 = a(e+2)^{2-1} = a(e+2) $. This is incorrect.

Since only option (A) yields the correct value $ ae^2 $ for the second term, it is the correct equation.

Question

The first term of an arithmetic sequence is $4$, and the common difference of the sequence is $3$. What is the seventh term of the sequence?

(A) $22$
(B) $25$
(C) $972$
(D) $2,916$

▶️ Answer/Explanation

Detailed solution

The formula for the $n$-th term of an arithmetic sequence is given by $a_n = a_1 + (n – 1)d$.
Here, the first term is $a_1 = 4$, the common difference is $d = 3$, and we need to find the seventh term ($n = 7$).
Substituting these values into the formula gives:
$a_7 = 4 + (7 – 1)3$
Simplify the expression inside the parentheses:
$a_7 = 4 + (6)3$
Multiply the terms:
$a_7 = 4 + 18$
Add to find the final result:
$a_7 = 22$

Question

A certain hobby shop wants to attract customers on slow sales days by offering a new discount. The shop offers a $1.5\%$ discount on Wednesdays with an added bonus for repeat customers. For customers who make purchases on successive Wednesdays, their discount is doubled from the previous Wednesday. For example, a certain customer makes purchases on three successive Wednesdays and receives $1.5\%$ (week $1$), $3\%$ (week $2$), and $6\%$ (week $3$) discounts on those three Wednesdays. Based on this pattern, what is the first successive week that a customer’s discount would exceed the purchase price of an item?

(A) $6$

(B) $7$

(D) $11$

▶️ Answer/Explanation

Detailed solution

The discount follows a geometric sequence where the first term $a_1 = 1.5\%$ and the common ratio $r = 2$.
Exceeding the purchase price means the discount percentage must be greater than $100\%$.
Week $1$: $1.5\%$
Week $2$: $3\%$
Week $3$: $6\%$
Week $4$: $12\%$
Week $5$: $24\%$
Week $6$: $48\%$
Week $7$: $96\%$
Week $8$: $192\%$, which is the first week to exceed $100\%$.
Thus, the correct option is (C).

Question

Fearing a new computer virus, an security company performs a simulation to predict the number of computers that might be affected by the virus. The number of infected computers can be modeled by a geometric sequence, where the first day of the simulation is day 1. On day 6, the virus had infected 750 computers, and on day 10 the virus had infected 3200 computers. To the nearest whole number, how many computers had been infected by the virus on day 14 based on the simulation?

(A) $5,650$

(B) $5,717$

(D) $120,329$

▶️ Answer/Explanation

Detailed solution

The number of infected computers follows a geometric sequence given by $a_n = a_1 \cdot r^{n-1}$.

We are provided with the values for day 6 and day 10: $a_6 = 750$ and $a_{10} = 3200$.

Using the property of geometric sequences, we can relate these terms: $a_{10} = a_6 \cdot r^{(10-6)} \Rightarrow 3200 = 750 \cdot r^4$.

Solving for the ratio factor $r^4$, we get $r^4 = \frac{3200}{750} = \frac{64}{15}$.

To find the number of infected computers on day 14 ($a_{14}$), we use the relationship $a_{14} = a_{10} \cdot r^{(14-10)} = a_{10} \cdot r^4$.

Substituting the known values: $a_{14} = 3200 \cdot \frac{64}{15} = \frac{204800}{15}$.

Calculating the final value: $a_{14} \approx 13653.33$.

Rounding to the nearest whole number, we get $13,653$.

Therefore, the correct option is (C).

Question

Values of the terms of an arithmetic sequence $a_n$ are graphed in the figure. Which of the following is an expression for the $n$th term of the arithmetic sequence?

(A) $a_n = -8 + 3(n – 1)$
(B) $a_n = -5 + 3(n – 1)$
(C) $a_n = -5 – 3(n – 1)$
(D) $a_n = -8 – 3(n – 1)$

▶️ Answer/Explanation

Detailed solution

Identify the first term $a_1$ from the graph at $n = 1$, which is $-5$.
Find the common difference $d$ by calculating the change between terms, such as $a_2 – a_1 = -2 – (-5) = 3$.
Recall the general formula for an arithmetic sequence: $a_n = a_1 + d(n – 1)$.
Substitute the identified values $a_1 = -5$ and $d = 3$ into the formula.
The resulting expression is $a_n = -5 + 3(n – 1)$.
Compare this result to the given options to find the match.
The correct option is (B).

Question (Calc Allowed)

The graph of a geometric sequence, $g_n$, is shown above. What is the value of $g_6$ ?

(A) $0.625$
(B) $0.3125$
(C) $0.15625$
(D) $0.078125$

▶️ Answer/Explanation

Detailed solution

Identify two consecutive terms from the graph: $g_2 = 5$ and $g_3 = \frac{5}{2}$.
Calculate the common ratio $r$ using $r = \frac{g_3}{g_2} = \frac{5/2}{5} = \frac{1}{2}$ or $0.5$.
The general formula for the $n^{th}$ term is $g_n = g_2 \cdot r^{n-2}$.
Substitute the values to find $g_6$: $g_6 = 5 \cdot (0.5)^{6-2}$.
Calculate the power: $g_6 = 5 \cdot (0.5)^4 = 5 \cdot 0.0625$.
Final calculation: $g_6 = 0.3125$.
The correct option is (B).

Question (Calc Allowed)

An arithmetic sequence has the form $a_n = a_k + d(n – k)$. The graph of an arithmetic sequence is shown above. Which of the following is the corresponding linear function?

(A) $f(x) = 5 + 2(x – 3)$
(B) $f(x) = 2 + 5(x – 3)$
(C) $f(x) = 5 – 2(x – 3)$
(D) $f(x) = 2 – 5(x – 3)$

▶️ Answer/Explanation

Detailed solution

Identify two points from the graph: $(1, 1)$ and $(2, 3)$.
Calculate the common difference (slope) using $d = \frac{a_2 – a_1}{2 – 1} = \frac{3 – 1}{1} = 2$.
Locate the value of the sequence at $k = 3$, which is the point $(3, 5)$, so $a_3 = 5$.
Substitute $a_k = 5$, $d = 2$, and $k = 3$ into the formula $f(x) = a_k + d(x – k)$.
The resulting linear function is $f(x) = 5 + 2(x – 3)$.
Comparing this to the given choices, the correct option is (A).

\(n\)	0	1	2	3	4
\(s_n\)	12	6	0	\(-6\)	\(-12\)

\(n\)	\(1\)	\(5\)	\(7\)	\(8\)	\(15\)
\(a_n\)	\(b\)	\(32\)	\(26\)	\(c\)	\(2\)

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