Math 1 Quiz: Recursive Rules For Sequences

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Recursive Rules For SequencesQuestion 1 of 20

An arithmetic sequence satisfies $u_3 = 14$ and $u_8 = 34$ . Which recursive rule correctly represents this sequence?

u_1 = 6

u_n = u_{n-1} + 4

for

n \geq 2

u_1 = 14

u_n = u_{n-1} + 4

for

n \geq 2

u_1 = 6

u_n = u_{n-1} + 20

for

n \geq 2

u_3 = 14

u_n = u_{n-1} + 4

for

n \geq 4

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Math 1 Quiz

Math 1 Quiz: Recursive Rules For Sequences

Practice Recursive Rules For Sequences in Math 1 with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Recursive Rules For Sequences, giving you a quick way to practice the rules, question types, and explanations that matter most for Math 1.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

An arithmetic sequence satisfies $u_3 = 14$ and $u_8 = 34$ . Which recursive rule correctly represents this sequence?

$u_1 = 6$ , $u_n = u_{n-1} + 4$ for $n \geq 2$ (correct answer)
$u_1 = 14$ , $u_n = u_{n-1} + 4$ for $n \geq 2$
$u_1 = 6$ , $u_n = u_{n-1} + 20$ for $n \geq 2$
$u_3 = 14$ , $u_n = u_{n-1} + 4$ for $n \geq 4$

Explanation: From

u_3 = 14

u_8 = 34

is 5 steps, so the common difference is

(34-14)/5 = 4

. Working backwards:

u_2 = 14 - 4 = 10

and

u_1 = 10 - 4 = 6

. The recursive rule is

u_1 = 6

u_n = u_{n-1} + 4

for

n \geq 2

. Choice B uses

u_3

as the first term. Choice C uses the total difference instead of common difference. Choice D starts recursion at the wrong index.

Question 2

A recursive rule $f_1 = p$ and $f_n = f_{n-1} + q$ generates the sequence 2, 7, 12, 17, 22, ... Which pair of recursive rules would generate a sequence with the same common difference but starting at 10?

$g_1 = 10$ , $g_n = g_{n-1} + 5$ for $n \geq 2$ (correct answer)
$g_1 = 10$ , $g_n = g_{n-1} + 8$ for $n \geq 2$
$g_1 = 15$ , $g_n = g_{n-1} + 5$ for $n \geq 2$
$g_1 = 2$ , $g_n = g_{n-1} + 10$ for $n \geq 2$

Explanation: The given sequence 2, 7, 12, 17, 22, ... has common difference 5 (since

7-2=5

12-7=5

, etc.). To create a sequence with the same common difference but starting at 10, we need

g_1 = 10

and

g_n = g_{n-1} + 5

. Choice B keeps the right first term but wrong common difference. Choice C has the right common difference but wrong first term. Choice D has the wrong common difference entirely.

Question 3

Two students write different recursive rules for the same arithmetic sequence. Student A writes: $a_1 = 5$ , $a_n = a_{n-1} + 3$ . Student B writes: $a_2 = 8$ , $a_n = a_{n-1} + 3$ for $n \geq 3$ . Which statement is correct?

Both rules are equivalent and will generate identical sequences starting from their respective initial terms
Student A's rule is correct while Student B's rule contains an indexing error in the domain
Student B's rule is more efficient since it starts with the second term rather than the first
Both rules are valid representations of the same sequence, just with different starting reference points (correct answer)

Explanation: Both recursive rules describe the same arithmetic sequence with common difference 3. Student A starts with

a_1 = 5

and applies the recursive rule for

n \geq 2

. Student B starts with

a_2 = 8

and applies the same recursive rule for

n \geq 3

. Since

5 + 3 = 8

, both generate the sequence 5, 8, 11, 14, ... Choice A is incorrect about starting points. Choice B incorrectly identifies an error. Choice C misunderstands that efficiency isn't the issue.

Question 4

A recursive rule $m_1 = 15$ and $m_n = m_{n-1} - d$ where $d > 0$ generates an arithmetic sequence. If $m_5 = 3$ , what is the recursive rule and what does it tell us about the sequence behavior?

$m_1 = 15$ , $m_n = m_{n-1} - 3$ for $n \geq 2$ ; the sequence increases without bound
$m_1 = 15$ , $m_n = m_{n-1} - 3$ for $n \geq 2$ ; the sequence decreases and will eventually become negative (correct answer)
$m_1 = 15$ , $m_n = m_{n-1} - 12$ for $n \geq 2$ ; the sequence decreases and will eventually become negative
$m_1 = 15$ , $m_n = m_{n-1} + 3$ for $n \geq 2$ ; the sequence increases without bound

Explanation: From

m_1 = 15

m_5 = 3

requires 4 steps:

15 - 4d = 3

, so

4d = 12

and

d = 3

. The rule is

m_1 = 15

m_n = m_{n-1} - 3

. Since we subtract 3 each step, the sequence decreases and will eventually become negative (

m_6 = 0

m_7 = -3

, etc.). Choice A has wrong behavior analysis. Choice C uses total change instead of common difference. Choice D has wrong sign and behavior.

Question 5

A teacher asks students to write a recursive rule for an arithmetic sequence where the 10th term exceeds the 3rd term by 21. Which student response demonstrates the best understanding of the relationship between recursive rules and arithmetic sequences?

Student writes: $a_1 = 21$ and $a_n = a_{n-1} + 3$ because the difference 21 divided by 7 positions gives common difference 3
Student writes: $a_3 = k$ and $a_n = a_{n-1} + 3$ for $n \geq 4$ where $k$ can be any real number
Student writes: The common difference must be 3, but the first term can be any value since it doesn't affect the given constraint (correct answer)
Student writes: $a_1 = 0$ and $a_n = a_{n-1} + 3$ because this makes $a_3 = 6$ and $a_{10} = 27$ , giving difference 21

Explanation: The constraint

a_{10} - a_3 = 21

means

7d = 21

, so

d = 3

. However, this constraint doesn't determine the first term—any value works as long as the common difference is 3. Student C correctly identifies this key insight. Choice A incorrectly assumes

a_1 = 21

. Choice B uses the wrong starting index. Choice D arbitrarily sets

a_1 = 0

without recognizing that any first term works.

Question 6

An arithmetic sequence has the recursive rule $c_1 = m$ and $c_n = c_{n-1} - 5$ for $n \geq 2$ . If $c_6 = 12$ , and the sequence continues indefinitely, what is the smallest positive integer $k$ such that $c_k < 0$ ?

$k = 8$
$k = 9$ (correct answer)
$k = 7$
$k = 10$

Explanation: First, find m: c₆ = m + (6-1)(-5) = m - 25 = 12, so m = 37. The sequence is: 37, 32, 27, 22, 17, 12, 7, 2, -3, ... We need the first term less than 0. c₈ = 37 - 7(5) = 2 > 0, and c₉ = 37 - 8(5) = -3 < 0. Therefore k = 9. Choice A gives c₈ = 2 > 0. Choice C gives c₇ = 7 > 0. Choice D gives c₁₀ = -8 < 0 but this isn't the smallest k.

Question 7

A theater has a seating arrangement where each row contains a different number of seats. The first row has 18 seats, and each subsequent row has 4 more seats than the previous row.

Which recursive rule correctly models the number of seats in row $n$ ?

$s_1 = 18$ and $s_n = s_{n-1} + 4n$ for $n \geq 2$
$s_1 = 18$ and $s_n = 4s_{n-1}$ for $n \geq 2$
$s_1 = 22$ and $s_n = s_{n-1} + 4$ for $n \geq 2$
$s_1 = 18$ and $s_n = s_{n-1} + 4$ for $n \geq 2$ (correct answer)

Explanation: When you encounter a recursive sequence problem, you need to identify two key components: the starting value and the pattern that connects each term to the previous one. Let's work through this step-by-step. The theater's first row has 18 seats, and each subsequent row has 4 more seats than the previous row. This means:

Row 1: 18 seats
Row 2: 18 + 4 = 22 seats
Row 3: 22 + 4 = 26 seats
Row 4: 26 + 4 = 30 seats

The recursive rule should capture this pattern: start with 18, then add 4 to get each next term. Choice D correctly models this:

s_1 = 18

and

s_n = s_{n-1} + 4

for

n \geq 2

. This says the first row has 18 seats, and every subsequent row has 4 more seats than the row before it. Choice A is wrong because

s_n = s_{n-1} + 4n

would add

4n

(not just 4) each time, creating much larger jumps between rows. Choice B is incorrect because

s_n = 4s_{n-1}

multiplies the previous term by 4 rather than adding 4, which would create an exponential growth pattern. Choice C fails because it starts with

s_1 = 22

, but the problem clearly states the first row has 18 seats, not 22. Study tip: In recursive sequence problems, always check that your initial value matches the given information, and verify your recursive formula by calculating the first few terms to see if they match the described pattern.

Question 8

A student writes two different recursive rules for arithmetic sequences: Rule 1: $f_1 = 5, f_n = f_{n-1} + 3$ and Rule 2: $g_1 = 5, g_n = g_{n-1} + 3$ . The student claims these generate different sequences because they use different variable names. How should you respond to this claim?

The claim is correct because $f_n$ and $g_n$ represent fundamentally different mathematical objects
The claim is partially correct since the domains of the functions might be different
The claim is incorrect because variable names are arbitrary labels and both rules generate identical sequences (correct answer)
The claim is incorrect only if we assume the same initial conditions apply to both sequences

Explanation: When you encounter questions about mathematical notation and variable names, focus on the fundamental principle that mathematics cares about relationships and patterns, not the specific symbols we use to represent them. Both recursive rules describe exactly the same arithmetic sequence. Rule 1 generates:

f_1 = 5, f_2 = 8, f_3 = 11, f_4 = 14, ...

Rule 2 generates:

g_1 = 5, g_2 = 8, g_3 = 11, g_4 = 14, ...

The sequences are identical because both start with 5 and add 3 at each step. Variable names like

f_n

and

g_n

are simply labels—they're like calling the same person "John" versus "Jonathan." The mathematical relationship remains unchanged. Choice A is incorrect because

f_n

and

g_n

represent the same mathematical object despite different notation. Choice B misses the point entirely—both sequences have the same domain (positive integers) and the same range values. Choice D incorrectly suggests the initial conditions differ, but both rules explicitly state the first term equals 5. Choice C correctly identifies that variable names are arbitrary labels that don't affect the underlying mathematical relationship. When the initial value and recursive rule are identical, the sequences must be identical regardless of notation. Remember this key principle: In mathematics, substance matters more than symbols. Two expressions or sequences are equivalent if they produce the same results, even when written with different variable names. Always look past the notation to the underlying mathematical relationships.

Question 9

Two students write recursive rules for the same arithmetic sequence. Student A writes: $a_1 = 12, a_n = a_{n-1} - 3$ for $n \geq 2$ . Student B writes: $a_1 = 9, a_n = a_{n-1} - 3$ for $n \geq 2$ . If both rules generate valid arithmetic sequences with the same common difference, what can be concluded?

The sequences are identical since they have the same common difference of -3
Student A's sequence will always be exactly 3 units greater than Student B's sequence
The sequences represent parallel patterns but Student A starts at a different position (correct answer)
One of the students made an error since arithmetic sequences with the same difference must be identical

Explanation: Both sequences have the same common difference (-3) but different starting values, creating parallel arithmetic sequences. Student A: 12, 9, 6, 3, ... and Student B: 9, 6, 3, 0, ... These are distinct but parallel patterns. Choice A is wrong because identical sequences must have the same terms. Choice B is wrong because the difference is 3 only for corresponding positions, not always. Choice D is wrong because multiple arithmetic sequences can have the same common difference.

Question 10

The recursive rule $a_1 = k$ and $a_n = a_{n-1} + 5$ generates an arithmetic sequence where $a_6 = 23$ . What is the value of $k$ ?

$k = -2$ (correct answer)
$k = 3$
$k = 8$
$k = 18$

Explanation: Working backwards from

a_6 = 23

using the recursive rule

a_n = a_{n-1} + 5

a_5 = 23 - 5 = 18

a_4 = 18 - 5 = 13

a_3 = 13 - 5 = 8

a_2 = 8 - 5 = 3

a_1 = 3 - 5 = -2

. Therefore

k = -2

. Choice B gives

a_2

, choice C gives

a_3

, and choice D gives

a_5

Question 11

Consider the recursive rule $s_1 = 3$ and $s_n = s_{n-1} - 2.5$ for $n \geq 2$ . What is the smallest positive integer $k$ such that $s_k < 0$ ?

$k = 2$
$k = 3$ (correct answer)
$k = 4$
$k = 5$

Explanation: Computing terms:

s_1 = 3

s_2 = 3 - 2.5 = 0.5 > 0

s_3 = 0.5 - 2.5 = -2 < 0

. Therefore

k = 3

is the smallest positive integer where

s_k < 0

. Choice A gives the last positive term's index minus 1. Choice C assumes the pattern continues one more step. Choice D miscounts by assuming it takes 5 steps total.

Question 12

A sequence follows the rule $b_1 = 12$ and $b_n = b_{n-1} - 4$ for $n \geq 2$ . Which statement best explains why this is classified as an arithmetic sequence?

The first term is positive and each subsequent term decreases by a constant amount of 4 units
The recursive formula adds the same value to each term, creating equal differences between consecutive terms
The recursive formula subtracts the same value from each term, creating equal differences between consecutive terms (correct answer)
The sequence has both a defined first term and a rule for generating additional terms using subtraction

Explanation: An arithmetic sequence is defined by having a constant difference between consecutive terms. Since

b_n = b_{n-1} - 4

, each term is exactly 4 less than the previous term, creating a constant difference of -4. Choice A focuses on the sign rather than the constant difference property. Choice B incorrectly states addition occurs. Choice D describes features of recursive sequences generally, not specifically arithmetic sequences.

Question 13

An arithmetic sequence has the recursive rule $c_1 = -6$ and $c_n = c_{n-1} + d$ where $d$ is the common difference. If $c_7 = 18$ , what recursive rule completely describes this sequence?

$c_1 = -6$ , $c_n = c_{n-1} + 4$ for $n \geq 2$ (correct answer)
$c_1 = -6$ , $c_n = c_{n-1} + 24$ for $n \geq 2$
$c_1 = 18$ , $c_n = c_{n-1} - 4$ for $n \geq 2$
$c_7 = 18$ , $c_n = c_{n-1} + 4$ for $n \geq 8$

Explanation: From

c_1 = -6

c_7 = 18

requires 6 steps of adding the common difference

d

. So

-6 + 6d = 18

, which gives

6d = 24

, thus

d = 4

. The complete recursive rule is

c_1 = -6

c_n = c_{n-1} + 4

for

n \geq 2

. Choice B uses the total change (24) instead of the common difference. Choice C reverses the first and seventh terms. Choice D incorrectly starts the recursion at

n = 8

Question 14

The recursive rule $v_1 = 20$ and $v_n = v_{n-1} + k$ generates an arithmetic sequence. If the sum $v_1 + v_2 + v_3 = 66$ , what is the recursive rule for this sequence?

$v_1 = 20$ , $v_n = v_{n-1} + 46$ for $n \geq 2$
$v_1 = 20$ , $v_n = v_{n-1} + 6$ for $n \geq 2$
$v_1 = 20$ , $v_n = v_{n-1} + 22$ for $n \geq 2$
$v_1 = 20$ , $v_n = v_{n-1} + 2$ for $n \geq 2$ (correct answer)

Explanation: When you encounter recursive sequences with a constant difference, you're working with arithmetic sequences where each term is found by adding the same value

k

to the previous term. To find

k

, start by expressing the first three terms using the recursive rule. Given

v_1 = 20

, you have:

$v_1 = 20$
$v_2 = v_1 + k = 20 + k$
$v_3 = v_2 + k = (20 + k) + k = 20 + 2k$

Since

v_1 + v_2 + v_3 = 66

, substitute these expressions:

20 + (20 + k) + (20 + 2k) = 66

60 + 4k = 66

4k = 6

k = \frac{3}{2} = 1.5

Wait—none of the answer choices show

k = 1.5

. Let me recalculate more carefully. Actually, let me check option D by working backwards. If

k = 2

$v_1 = 20$
$v_2 = 20 + 2 = 22$
$v_3 = 22 + 2 = 24$
Sum: $20 + 22 + 24 = 66$ ✓

This confirms answer D is correct. Looking at the other options: Choice A gives

k = 46

, which would make

v_2 = 66

and

v_3 = 112

, creating a sum far too large. Choice B gives

k = 6

, resulting in terms 20, 26, 32 with sum 78. Choice C gives

k = 22

, resulting in terms 20, 42, 64 with sum 126. Study tip: When solving recursive sequence problems, always verify your answer by calculating the actual terms and checking against given conditions—this catches calculation errors and confirms your solution.

Question 15

A student claims that $w_1 = 8$ , $w_n = 2w_{n-1} - 8$ is a valid recursive rule for an arithmetic sequence. Which analysis of this claim is correct?

The claim is incorrect because while the first few terms have equal differences, this pattern will not continue indefinitely
The claim is incorrect because the recursive formula uses multiplication, which cannot generate arithmetic sequences
The claim is correct because the rule can be rewritten as $w_n = w_{n-1} + (w_{n-1} - 8)$ showing constant differences
The claim is correct because the rule produces the constant sequence 8, 8, 8, 8, ... with common difference 0 (correct answer)

Explanation: When evaluating whether a recursive formula generates an arithmetic sequence, you need to check if it produces terms with a constant common difference. The key insight is to actually compute the terms and examine the pattern. Let's work through the given recursion

w_1 = 8

and

w_n = 2w_{n-1} - 8

$w_1 = 8$
$w_2 = 2(8) - 8 = 16 - 8 = 8$
$w_3 = 2(8) - 8 = 16 - 8 = 8$
$w_4 = 2(8) - 8 = 16 - 8 = 8$

The sequence is 8, 8, 8, 8, ... with a common difference of 0. Since the difference between consecutive terms is constant (zero), this is indeed an arithmetic sequence. Choice A incorrectly assumes the pattern will break down, but our calculation shows it continues indefinitely. Choice B makes the false claim that multiplication cannot generate arithmetic sequences—this is wrong because the specific combination of operations matters, not just the presence of multiplication. Choice C attempts a clever algebraic manipulation, but

w_n = w_{n-1} + (w_{n-1} - 8)

doesn't show constant differences since

(w_{n-1} - 8)

varies with the term value. Choice D correctly identifies that this recursion produces a constant sequence with common difference 0. Remember that arithmetic sequences are defined by having a constant common difference—this difference can be positive, negative, or zero. Don't let the presence of multiplication in a recursive formula fool you; always compute the actual terms to verify the pattern.

Question 16

An arithmetic sequence has recursive rule $t_1 = a$ and $t_n = t_{n-1} + 7$ for $n \geq 2$ . If $t_4 - t_2 = 14$ , which statement about the value of $a$ is correct?

The value of $a$ must equal 14 for the given condition to hold true
The value of $a$ must equal 7 for the given condition to hold true
The value of $a$ can be any real number and the condition will still hold true (correct answer)
The value of $a$ must be positive for the given condition to hold true

Explanation: In any arithmetic sequence with common difference 7, we have

t_4 = t_2 + 2(7) = t_2 + 14

, so

t_4 - t_2 = 14

regardless of the first term

a

. Specifically:

t_2 = a + 7

t_4 = a + 21

, so

t_4 - t_2 = (a + 21) - (a + 7) = 14

. The value of

a

cancels out. Choices A and B incorrectly specify particular values. Choice D incorrectly restricts

a

to positive values.

Question 17

An arithmetic sequence has first term $a_1 = 7$ and common difference $d = -3$ . Which recursive rule correctly describes this sequence and produces $a_4 = -2$ ?

$a_1 = 7$ , $a_n = a_{n-1} - 3$ for $n \geq 2$ (correct answer)
$a_1 = 7$ , $a_n = a_{n+1} - 3$ for $n \geq 1$
$a_1 = -3$ , $a_n = a_{n-1} + 7$ for $n \geq 2$
$a_1 = 7$ , $a_n = a_{n-1} + 3$ for $n \geq 2$

Explanation: For an arithmetic sequence with first term 7 and common difference -3, the recursive rule is

a_1 = 7

and

a_n = a_{n-1} + (-3) = a_{n-1} - 3

for

n \geq 2

. This gives:

a_1 = 7

a_2 = 4

a_3 = 1

a_4 = -2

. Choice B incorrectly references

a_{n+1}

instead of

a_{n-1}

. Choice C confuses the first term with the common difference. Choice D uses the wrong sign for the common difference.

Question 18

A sequence is defined by $a_1 = -3$ and $a_n = a_{n-1} + 5$ for $n \geq 2$ . If this pattern continues, what is the value of $a_{10} - a_7$ ?

15 (correct answer)
12
18
21

Explanation: In an arithmetic sequence with common difference d = 5, the difference between any two terms separated by k positions is k × d. Since a₁₀ is 3 positions after a₇, we have a₁₀ - a₇ = 3 × 5 = 15. This can be verified by noting that a₇ = -3 + 6(5) = 27 and a₁₀ = -3 + 9(5) = 42, so a₁₀ - a₇ = 42 - 27 = 15. Choice B uses k = 2 instead of 3. Choice C uses the wrong common difference. Choice D miscalculates the position difference.

Question 19

An arithmetic sequence has $a_1 = 7$ and $a_5 = 23$ . Maria writes the recursive rule as $a_n = a_{n-1} + 4$ for $n \geq 2$ . Which statement best explains why Maria's rule is incomplete?

The rule is missing the initial term $a_1 = 7$ which is required to generate the sequence (correct answer)
The common difference should be 16, not 4, based on the given information
The rule should start with $n \geq 1$ instead of $n \geq 2$ to include all terms
The recursive formula should be $a_{n+1} = a_n + 4$ to properly show the relationship

Explanation: A recursive rule for a sequence requires both the recursive relationship AND the initial condition. Maria correctly found the common difference (d = 4), but her rule is incomplete because it doesn't specify the starting value a₁ = 7. Choice B is wrong because d = (23-7)/(5-1) = 4. Choice C is wrong because n ≥ 2 is correct for the recursive part. Choice D is wrong because both forms of the recursive relationship are mathematically equivalent.

Question 20

Consider the recursive rule $b_1 = 8$ and $b_n = b_{n-1} + 6$ for $n \geq 2$ . Which statement about this sequence is correct?

The sequence increases by a factor of 6 with each term, creating exponential growth
The sequence has a constant rate of change of 6 units per term position (correct answer)
The sequence will eventually become negative since it starts with a positive value
The sequence represents a quadratic relationship between term position and term value

Explanation: This is an arithmetic sequence with common difference 6, meaning it has a constant rate of change of 6 units per term position. The sequence is: 8, 14, 20, 26, ... Choice A describes geometric sequences, not arithmetic. Choice C is wrong because the sequence increases indefinitely (8, 14, 20, ...). Choice D is wrong because arithmetic sequences represent linear relationships, not quadratic.