Discrete Math Quiz: Checking Solutions

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Checking SolutionsQuestion 1 of 20

A student claims that the function $f(x) = \lfloor x \rfloor + \lfloor -x \rfloor$ equals zero for all real numbers $x$ . To validate this reasoning, which of the following approaches would be most appropriate?

Check the formula for several integer values and conclude it holds for all real numbers

Verify the claim by testing both integer and non-integer values, noting that the function equals

-1

when

x

is not an integer

Use the property that

\lfloor -x \rfloor = -\lfloor x \rfloor

to algebraically prove the function equals zero

Apply the definition of floor function only to positive values since negative inputs complicate the analysis

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Discrete Math Quiz

Discrete Math Quiz: Checking Solutions

Practice Checking Solutions in Discrete Math 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 Checking Solutions, giving you a quick way to practice the rules, question types, and explanations that matter most for Discrete Math.

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

Check the formula for several integer values and conclude it holds for all real numbers
Verify the claim by testing both integer and non-integer values, noting that the function equals $-1$ when $x$ is not an integer (correct answer)
Use the property that $\lfloor -x \rfloor = -\lfloor x \rfloor$ to algebraically prove the function equals zero
Apply the definition of floor function only to positive values since negative inputs complicate the analysis

Explanation: To validate the student's claim, we must test both integer and non-integer cases. For integers:

f(2) = \lfloor 2 \rfloor + \lfloor -2 \rfloor = 2 + (-2) = 0

. For non-integers:

f(1.5) = \lfloor 1.5 \rfloor + \lfloor -1.5 \rfloor = 1 + (-2) = -1

. The claim is false. Choice A is insufficient (only tests integers). Choice C uses an incorrect property (

\lfloor -x \rfloor \neq -\lfloor x \rfloor

in general). Choice D ignores necessary test cases.

Question 2

A student solving a system of modular equations finds $x \equiv 17 \pmod{60}$ as the solution to the system $x \equiv 5 \pmod{12}$ and $x \equiv 2 \pmod{5}$ . To validate this solution, which verification approach is most thorough?

Substitute $x = 17$ into both original congruences and verify they hold true
Check that $\gcd(12, 5) = 1$ and apply the Chinese Remainder Theorem formula
Verify that $x = 17$ satisfies the congruences and confirm the solution is unique modulo 60 (correct answer)
Calculate several values $x = 17, 77, 137$ and verify each satisfies both congruences

Explanation: Complete validation requires both checking that the proposed solution works and confirming it represents the complete solution set. Since

\gcd(12,5) = 1

, the Chinese Remainder Theorem guarantees a unique solution modulo

\text{lcm}(12,5) = 60

. Choice A only verifies the specific value works but doesn't confirm completeness. Choice B verifies the theorem applies but doesn't check the actual solution. Choice D unnecessarily checks multiple equivalent values when one verification plus uniqueness confirmation suffices.

Question 3

A proposed algorithm for finding the shortest path in a weighted graph claims to work by repeatedly selecting the edge with minimum weight that doesn't create a cycle. A colleague argues this is incorrect because it might miss the optimal solution. What is the most effective way to check whether the colleague's criticism is valid?

Test the algorithm on several random graphs and compare results with Dijkstra's algorithm
Prove mathematically that the greedy choice property holds for shortest path problems
Construct a specific weighted graph where the algorithm produces a suboptimal path (correct answer)
Verify that the algorithm correctly handles negative edge weights in all test cases

Explanation: The described algorithm is actually finding a minimum spanning tree, not shortest paths between vertices. The most definitive way to validate the colleague's criticism is to construct a counterexample where this approach fails for shortest path finding. For instance, in a triangle graph with edges of weights 1, 1, and 10, the algorithm might select both weight-1 edges for the MST, but the shortest path between the endpoints of the weight-10 edge is 2, not 1. Choice A might work but isn't definitive, choice B is investigating the wrong property, and choice D addresses an irrelevant issue since the fundamental approach is flawed regardless of edge weight signs.

Question 4

A solution to a recurrence relation problem states that $a_n = 3 \cdot 2^n - 2 \cdot 3^n$ satisfies the recurrence $a_n = 5a_{n-1} - 6a_{n-2}$ with initial conditions $a_0 = 1, a_1 = 0$ . To thoroughly validate this solution, which verification steps are both necessary and sufficient?

Check that the proposed solution satisfies the initial conditions and verify the characteristic equation
Substitute the solution into the recurrence relation and confirm it holds for all $n \geq 2$
Verify the initial conditions, substitute into the recurrence relation, and check uniqueness of the solution (correct answer)
Calculate $a_2, a_3, a_4$ using both the recurrence and the proposed formula and compare results

Explanation: Complete validation requires three components: (1) verifying initial conditions are satisfied, (2) confirming the proposed solution satisfies the recurrence relation for all valid n, and (3) establishing that the solution is unique (which follows from the theory of linear recurrences with constant coefficients). Choice A misses verification of the recurrence relation itself. Choice B misses initial conditions and uniqueness. Choice D only provides finite verification, which is insufficient for proving the solution works for all n. Only choice C encompasses all necessary verification steps.

Question 5

A student models a tournament bracket as a binary tree and claims that a tournament with $n = 2^k$ players requires exactly $n-1$ games. The reasoning states: "Each game eliminates one player, and we start with $n$ players and end with 1 winner, so $n-1$ eliminations are needed." How should this reasoning be validated?

Verify the tree structure has the correct number of internal nodes and check that each internal node represents exactly one game
Confirm that the elimination argument is sound and verify the formula works for several tournament sizes
Check that the binary tree model accurately represents the tournament structure and validate the counting argument (correct answer)
Prove by induction that a complete binary tree with $n$ leaves has $n-1$ internal nodes

Explanation: Validation should verify both that the binary tree model correctly represents the tournament structure (each internal node is a game, leaves are final rankings, tree is complete) and that the elimination counting argument is logically sound. Choice A focuses only on the tree properties without validating the model itself. Choice B checks the logic and examples but doesn't verify the model representation. Choice D proves a related mathematical fact but doesn't validate whether the model correctly represents the tournament problem.

Question 6

A Boolean logic solution claims that the expression $(p \land q) \lor (\neg p \land r) \lor (q \land r)$ is equivalent to $(p \land q) \lor (\neg p \land r)$ . To check this claimed equivalence, which approach would be most reliable?

Construct truth tables for both expressions and verify they have identical output columns (correct answer)
Use algebraic manipulation with Boolean laws to derive one expression from the other
Test the expressions with several representative combinations of truth values for $p$ , $q$ , and $r$
Apply De Morgan's laws to convert both expressions to disjunctive normal form and compare

Explanation: Truth tables provide a complete and systematic verification by exhaustively checking all possible truth value assignments. This is the most reliable method for Boolean equivalence because it's impossible for equivalent expressions to differ on any input. Choice B would work if executed correctly but is more prone to algebraic errors. Choice C provides only partial verification and might miss cases where the expressions differ. Choice D misapplies De Morgan's laws, which are used for negation of expressions, not for this type of comparison.

Question 7

A student claims that for any graph $G$ with $n$ vertices and $m$ edges, if $m > \binom{n-1}{2}$ , then $G$ must be connected. To validate this reasoning, which approach would definitively establish whether the claim is correct?

Construct a counterexample with $n = 5$ vertices, $m = 7$ edges, where the graph is disconnected (correct answer)
Prove that any graph with more than $\binom{n-1}{2}$ edges contains a spanning tree of size $n-1$
Show that the maximum number of edges in a disconnected graph on $n$ vertices is $\binom{n-1}{2}$
Verify the claim holds for small values $n = 3, 4, 5$ and conclude by mathematical induction

Explanation: To validate reasoning about a universal claim, finding a single counterexample definitively disproves it. A graph with 5 vertices where 4 vertices form a complete subgraph (6 edges) and 1 vertex is isolated, plus 1 additional edge between two vertices in the complete subgraph, gives 7 edges but remains disconnected. Since

\binom{4}{2} = 6 < 7

, this counterexample shows the claim is false. Choice B assumes the conclusion, choice C would actually support the claim if proven, and choice D cannot establish a universal claim through finite verification.

Question 8

A proposed solution to finding the number of spanning trees in a complete graph $K_n$ uses Kirchhoff's matrix-tree theorem and concludes the answer is $n^{n-2}$ . To validate this result, a student should verify that:

The Laplacian matrix was constructed correctly and the determinant calculation was performed accurately
The result matches Cayley's formula and gives correct values for small cases like $K_3$ and $K_4$
Both the matrix construction and determinant calculation are correct, and the result agrees with known formulas (correct answer)
The matrix-tree theorem applies to complete graphs and the resulting matrix has the expected eigenvalues

Explanation: Complete validation requires verifying the computational steps (correct Laplacian matrix construction and determinant calculation) and confirming the result matches established theory (Cayley's formula). This combines procedural verification with theoretical consistency checking. Choice A only covers the computational aspect. Choice B only covers the theoretical consistency. Choice D focuses on applicability and eigenvalues, which don't directly validate the specific calculation performed.

Question 9

A student claims that the generating function $G(x) = \frac{1}{(1-x)^3}$ counts the number of ways to distribute $n$ identical objects into 3 distinct bins. To validate this claim, what should be verified?

Check that the generating function has the correct form for problems involving identical objects and distinct containers
Confirm that $G(x) = (1 + x + x^2 + \cdots)^3$ and that each factor represents choices for one bin
Verify that $G(1) = \infty$ as expected for an infinite number of total distributions
Check that the coefficient of $x^n$ in the expansion equals $\binom{n+2}{2}$ and verify this matches the stars-and-bars formula (correct answer)

Explanation: When validating a generating function claim, you need to verify that the mathematical expression actually produces the correct counts for the stated combinatorial problem. The correct approach is D because it provides complete verification through two essential steps. First, you must confirm that the coefficient of

x^n

\frac{1}{(1-x)^3}

equals

\binom{n+2}{2}

. Using the generalized binomial theorem, this expansion gives coefficients

\binom{n+3-1}{3-1} = \binom{n+2}{2}

. Second, you verify this matches the stars-and-bars formula for distributing

n

identical objects into 3 distinct bins, which is indeed

\binom{n+3-1}{3-1} = \binom{n+2}{2}

. A is insufficient because checking the "correct form" doesn't verify the actual counts. Many problems involve identical objects and distinct containers but have different generating functions. B contains a fundamental error. The expression

(1 + x + x^2 + \cdots)^3

would count ways to place distinguishable objects (one per bin per term), not identical objects. While

\frac{1}{(1-x)^3}

does equal this expression, the interpretation is wrong for this problem. C is meaningless as verification. That

G(1) = \infty

simply reflects that there are infinitely many possible values of

n

, but tells you nothing about whether the coefficients count the right thing. Key strategy: Always verify generating functions by checking that specific coefficients match known combinatorial formulas. Abstract properties like convergence behavior rarely validate counting claims.

Question 10

A proof that a relation $R$ on set $S = \{a, b, c, d\}$ is an equivalence relation shows that $R$ is reflexive by verifying $(a,a), (b,b), (c,c), (d,d) \in R$ , and symmetric by checking that whenever $(x,y) \in R$ , then $(y,x) \in R$ for all pairs in a given list. To complete the validation, what must be verified for transitivity?

For each element $x \in S$ , verify that if $(x,y) \in R$ and $(y,z) \in R$ , then $(x,z) \in R$
Check that for all triples $(x,y,z)$ where $(x,y) \in R$ and $(y,z) \in R$ , we have $(x,z) \in R$ (correct answer)
Verify that the relation contains all pairs $(x,z)$ where there exists some $y$ with $(x,y), (y,z) \in R$
Confirm that for any chain $x_1, x_2, \ldots, x_n$ in $R$ , the pair $(x_1, x_n)$ is also in $R$

Explanation: Transitivity requires checking every possible triple of elements where the first two pairs are in R, then verifying the third pair is also in R. Choice B correctly states this comprehensive requirement. Choice A is incomplete because it suggests checking only for each starting element x, potentially missing some triples. Choice C restates the requirement but less precisely. Choice D describes a stronger condition (transitivity over arbitrary-length chains) which, while related, goes beyond the basic transitivity requirement and would be unnecessarily complex to verify directly.

Question 11

In a probability model for a discrete random process, a student calculates that events $A$ , $B$ , and $C$ have probabilities $P(A) = 0.4$ , $P(B) = 0.3$ , $P(C) = 0.5$ , with $P(A \cap B) = 0.1$ , $P(A \cap C) = 0.2$ , $P(B \cap C) = 0.15$ , and $P(A \cap B \cap C) = 0.05$ . What should be checked to validate this probability model?

Verify that each pairwise intersection probability is less than both individual event probabilities
Check that $P(A) + P(B) + P(C) = 1.2 > 1$ is acceptable since the events are not mutually exclusive
Confirm that the inclusion-exclusion principle gives a valid probability for $P(A \cup B \cup C)$
Verify that all individual and joint probabilities are between 0 and 1, and that $P(A \cup B \cup C) \leq 1$ (correct answer)

Explanation: When validating any probability model, you need to ensure it satisfies the fundamental axioms of probability theory. This means checking that all probabilities are non-negative, no probability exceeds 1, and the total probability of the sample space doesn't exceed 1. Let's verify this model using the inclusion-exclusion principle:

P(A \cup B \cup C) = P(A) + P(B) + P(C) - P(A \cap B) - P(A \cap C) - P(B \cap C) + P(A \cap B \cap C)

Substituting:

P(A \cup B \cup C) = 0.4 + 0.3 + 0.5 - 0.1 - 0.2 - 0.15 + 0.05 = 0.8

Since

P(A \cup B \cup C) = 0.8 \leq 1

, this model is valid. Answer D correctly identifies both necessary checks: all probabilities must be between 0 and 1 (they are), and

P(A \cup B \cup C) \leq 1

(it is). Answer A is incomplete—while pairwise intersections should be less than individual probabilities (and they are here), this alone doesn't validate the entire model. Answer B shows a fundamental misunderstanding: the sum

P(A) + P(B) + P(C) = 1.2

is irrelevant for non-mutually exclusive events, and this sum can exceed 1 without invalidating the model. Answer C is partially correct but incomplete—checking inclusion-exclusion gives a valid result, but you must also verify that this result doesn't exceed 1. Remember: probability model validation requires checking that all basic axioms hold. Always verify individual probabilities are in [0,1] and that the union of all events has probability ≤ 1.

Question 12

A student proves that a particular graph $G$ is Eulerian by showing it is connected and that every vertex has even degree. However, the student's connection proof only demonstrates that there is a path between vertices $v_1$ and $v_7$ . What error in reasoning needs to be addressed?

The student must verify that every vertex has degree exactly 2, not just even degree
Showing a path between two specific vertices is insufficient to prove the entire graph is connected (correct answer)
The student should have proven the graph is Hamiltonian rather than Eulerian for this application
Even degree is necessary but not sufficient; the student must also verify the graph is simple

Explanation: To prove a graph is connected, one must show that every pair of vertices has a path between them, not just one specific pair. The student's proof is incomplete because demonstrating connectivity between

v_1

and

v_7

doesn't establish that all other vertices are reachable from these or from each other. Choice A is incorrect because Eulerian graphs require even degree, not degree 2. Choice C misidentifies the theorem being applied. Choice D is wrong because the condition for Eulerian graphs doesn't require the graph to be simple (no multiple edges or loops).

Question 13

A combinatorial argument claims that the number of ways to distribute $n$ identical balls into $k$ distinct boxes is $\binom{n+k-1}{k-1}$ . A student verifies this by checking that when $n = 3$ and $k = 2$ , the formula gives $\binom{4}{1} = 4$ , and by listing the distributions: (3,0), (2,1), (1,2), (0,3). What additional validation should be performed?

Verify the formula for several other small values and check that the combinatorial interpretation is correct (correct answer)
Prove that the formula satisfies the recurrence relation for stars and bars problems
Confirm that the formula gives integer results for all positive values of $n$ and $k$
Test the formula on edge cases like $n = 0$ or $k = 1$ and verify boundary conditions

Explanation: While the student correctly verified one case, additional validation should include testing more cases and ensuring the combinatorial interpretation (stars and bars method) correctly applies to the problem setup. The student should verify that the problem indeed involves identical balls and distinct boxes, and that empty boxes are allowed. Choice B involves proving the underlying theory rather than validating the application. Choice C is automatically satisfied since binomial coefficients are always integers for valid inputs. Choice D tests edge cases but doesn't address whether the fundamental model is correct for the stated problem.

Question 14

A proof by induction claims that $\sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}$ for all positive integers $n$ . In the inductive step, a student writes: "Assume the formula holds for $k$ . Then $\sum_{i=1}^{k+1} i^2 = \sum_{i=1}^k i^2 + (k+1)^2 = \frac{k(k+1)(2k+1)}{6} + (k+1)^2$ ." What additional verification is needed to complete the proof?

Show that the base case $n = 1$ satisfies the formula and verify the inductive hypothesis is correctly stated
Algebraically simplify $\frac{k(k+1)(2k+1)}{6} + (k+1)^2$ to obtain $\frac{(k+1)(k+2)(2k+3)}{6}$ (correct answer)
Verify that $\sum_{i=1}^0 i^2 = 0$ and confirm the formula works for several small values of $n$
Prove that the formula is unique by showing no other polynomial expression could satisfy the same recurrence

Explanation: The student has correctly set up the inductive step but hasn't completed the crucial algebraic manipulation to show that the expression equals the formula for

n = k+1

. The inductive step requires proving that

\frac{k(k+1)(2k+1)}{6} + (k+1)^2 = \frac{(k+1)(k+2)(2k+3)}{6}

. Choice A addresses the base case, which is separate from completing this inductive step. Choice C suggests unnecessary verification. Choice D is irrelevant to completing the induction proof. Only choice B identifies the specific algebraic work needed to finish the inductive step.

Question 15

A student models a scheduling problem as a graph coloring problem where courses are vertices and edges connect courses that cannot be scheduled simultaneously. The student claims that if the graph has chromatic number $\chi(G) = 4$ , then exactly 4 time slots are needed. To validate this reasoning, what potential flaw should be investigated?

Whether the graph construction correctly represents all scheduling constraints from the original problem (correct answer)
Whether the chromatic number calculation used the most efficient algorithm available
Whether the graph is planar, since non-planar graphs may have different coloring properties
Whether the student considered that larger graphs typically require more computational time to color

Explanation: The critical flaw to investigate is whether the graph model accurately captures all constraints from the real scheduling problem. If the student missed constraints (like room capacity, instructor availability, or prerequisite requirements), the graph might be incomplete, leading to an underestimate of required time slots. The chromatic number gives the minimum colors needed for the modeled graph, but if the model is incomplete, this doesn't reflect the actual scheduling requirements. Choices B and D relate to computational efficiency but don't affect correctness. Choice C addresses graph properties that don't directly impact the validity of the chromatic number calculation for scheduling.

Question 16

To verify that the recurrence relation $a_n = 3a_{n-1} - 2a_{n-2}$ with $a_0 = 1, a_1 = 2$ has the closed form $a_n = 2^n + (-1)^n \cdot 1^n$ , a student computes $a_2$ using both methods. Which calculation correctly validates the proposed solution?

Recurrence: $a_2 = 3(2) - 2(1) = 4$ ; Closed form: $a_2 = 2^2 + (-1)^2 \cdot 1^2 = 4 + 1 = 5$ ; Solution invalid (correct answer)
Recurrence: $a_2 = 3(2) - 2(1) = 4$ ; Closed form: $a_2 = 2^2 + (-1)^2 \cdot 1^2 = 4 + 1 = 5$ ; Solution valid since both are positive
Recurrence: $a_2 = 3(2) - 2(1) = 4$ ; Closed form: $a_2 = 2^2 + (-1)^2 \cdot 1^2 = 4 + 1 = 5$ ; Need more terms to conclude
Recurrence: $a_2 = 3(1) - 2(2) = -1$ ; Closed form: $a_2 = 2^2 + (-1)^2 \cdot 1^2 = 4 + 1 = 5$ ; Solution invalid

Explanation: Using the recurrence:

a_2 = 3a_1 - 2a_0 = 3(2) - 2(1) = 6 - 2 = 4

. Using the proposed closed form:

a_2 = 2^2 + (-1)^2 \cdot 1^2 = 4 + 1 = 5

. Since

4 \neq 5

, the proposed solution is invalid. Choice B incorrectly concludes validity despite different values. Choice C suggests more testing when one counterexample suffices. Choice D has an arithmetic error in the recurrence calculation.

Question 17

A student uses strong induction to prove that every integer $n \geq 2$ can be written as a sum of distinct powers of 2. The base cases check $n = 2 = 2^1$ and $n = 3 = 2^1 + 2^0$ . For the inductive step, which reasoning best validates the logical structure?

Assume the statement holds for all $k < n$ , then show $n = 2^j + m$ where $m < n$ and $2^j$ doesn't appear in $m$ 's representation (correct answer)
The binary representation of $n$ directly gives the required sum, making induction unnecessary for this problem
Assume the statement holds for $n-1$ , then add 1 to get a representation for $n$ using distinct powers
Check that if $n$ is even, use $n/2$ 's representation shifted by one power; if odd, subtract 1 and apply even case

Explanation: Strong induction requires assuming the statement for all

k < n

, then proving for

n

. The correct approach: let

2^j

be the largest power of 2 with

2^j \leq n

. Then

n = 2^j + m

where

0 \leq m < 2^j < n

. By the inductive hypothesis,

m

has a representation using distinct powers of 2, all smaller than

2^j

, so

2^j

doesn't conflict. Choice B abandons induction entirely. Choice C uses weak induction (insufficient). Choice D gives a specific algorithm but doesn't follow strong induction structure.

Question 18

A student models a scheduling problem using graph coloring and claims that the chromatic number of their conflict graph is 4, meaning 4 time slots are needed. To validate this solution approach, which verification would be most comprehensive?

Confirm the graph has maximum degree 3, since chromatic number ≤ maximum degree + 1 by Brook's theorem
Verify the graph model correctly represents conflicts, then check that 4 colors suffice for the proposed coloring
Calculate the clique number and ensure it doesn't exceed 4, since chromatic number ≥ clique number
Check that a 4-coloring exists by verifying no adjacent vertices share colors, then confirm no 3-coloring is possible (correct answer)

Explanation: When validating a graph coloring solution for scheduling problems, you need to verify both that your proposed solution works and that it's optimal (uses the minimum number of colors). The correct approach is D because comprehensive verification requires two steps: first confirming that your 4-coloring is valid (no adjacent vertices share the same color), then proving optimality by showing no 3-coloring exists. Only by demonstrating that fewer colors are impossible can you confidently claim the chromatic number is exactly 4. Let's examine why the other options fall short. Option A uses Brook's theorem incorrectly - while the chromatic number is at most

\Delta + 1

(where

\Delta

is maximum degree), having maximum degree 3 only tells you the chromatic number is at most 4, not that it equals 4. Option B focuses on model validation and checking the 4-coloring works, but crucially omits verifying that 4 is the minimum number needed. Option C uses the clique number bound correctly (chromatic number ≥ clique number), but this only provides a lower bound - finding a clique of size 3 doesn't prove you need exactly 4 colors. The key insight is that determining chromatic number requires proving both an upper bound (showing

k

colors suffice) and a lower bound (showing

k-1

colors don't suffice). Only option D addresses both requirements. Remember: when verifying chromatic number claims, always check both directions - that your proposed coloring works AND that fewer colors fail. This two-part verification is essential for optimization problems disguised as graph theory.

Question 19

A student proves that $\sum_{k=1}^{n} k^2 = \frac{n(n+1)(2n+1)}{6}$ by induction and then claims this validates that $\sum_{k=1}^{100} k^2 = 338350$ . To check this calculation, which approach identifies the error?

Verify the induction proof is correct, then accept the numerical result since the formula is proven valid
Substitute $n = 100$ into the proven formula: $\frac{100 \cdot 101 \cdot 201}{6} = \frac{2,030,100}{6} = 338,350$ , confirming the result
Check boundary cases like $n = 1, 2, 3$ to verify the formula works before trusting the $n = 100$ calculation
Compute $\frac{100 \cdot 101 \cdot 201}{6} = \frac{2,030,100}{6} \approx 338,350$ , but verify that 6 divides 2,030,100 exactly (correct answer)

Explanation: Even with a correct formula, computational errors can occur. We must verify:

\frac{100 \cdot 101 \cdot 201}{6} = \frac{2,030,100}{6}

. Since

2,030,100 = 6 \times 338,350

, the division is exact and the answer is correct. Choice A skips verification of the specific calculation. Choice B shows the work but doesn't emphasize checking for exact division. Choice C tests the wrong aspect (formula validity rather than computational accuracy). Choice D properly validates both the arithmetic and exactness of division.

Question 20

A student models a population growth problem using the recurrence $P_{n+1} = 1.2P_n - 100$ with $P_0 = 1000$ , claiming the population stabilizes at 500. To validate this equilibrium analysis, which verification is most thorough?

Solve $P = 1.2P - 100$ to get $P = 500$ , confirming the equilibrium exists mathematically
Check that $P_1 = 1.2(1000) - 100 = 1100 > 1000$ , so the population grows toward the equilibrium
Verify equilibrium exists at $P = 500$ , then check stability by testing $P_{n+1} - 500 = 1.2(P_n - 500)$ shows convergence (correct answer)
Calculate several terms: $P_1 = 1100, P_2 = 1220, P_3 = 1364$ to confirm the sequence approaches 500

Explanation: Complete validation requires both existence and stability analysis. Setting

P_{n+1} = P_n = P

gives

P = 1.2P - 100

, so

P = 500

. For stability, let

u_n = P_n - 500

. Then

u_{n+1} = P_{n+1} - 500 = 1.2P_n - 100 - 500 = 1.2(P_n - 500) = 1.2u_n

. Since

|1.2| > 1

, the equilibrium is unstable and the population diverges. Choice A only checks existence. Choice B and D show growth but don't analyze long-term behavior. Choice C correctly identifies that the equilibrium exists but is unstable.