An automated data-processing system updates two state metrics, X and Y, through a sequenti… — GMAT

An automated data-processing system updates two state metrics, $X$ and $Y$ , through a sequential three-stage pipeline ( $k = 1, 2, 3$ ). Starting from an initial state $(X_0, Y_0)$ , the system updates the state variables at each stage $k$ according to the following transition rules:

1. $X_k = X_{k-1} + 2Y_{k-1}$
2. If $X_{k-1}$ is even, $Y_k = Y_{k-1} + k$ ; if $X_{k-1}$ is odd, $Y_k = Y_{k-1} - k$ .

Match each initial state configuration $(X_0, Y_0)$ on the left to its corresponding final state $(X_3, Y_3)$ after Stage 3 on the right.

Initial State: $(X_0, Y_0) = (2, 3)$ Final State: $(X_3, Y_3) = (28, 9)$
Initial State: $(X_0, Y_0) = (3, 5)$ Final State: $(X_3, Y_3) = (25, -1)$
Initial State: $(X_0, Y_0) = (4, 1)$ Final State: $(X_3, Y_3) = (18, 7)$
Initial State: $(X_0, Y_0) = (5, 2)$ Final State: $(X_3, Y_3) = (9, -4)$

Answer

Initial State

(2,3)

matches Final State

(28,9)

; Initial State

(3,5)

matches Final State

(25,-1)

; Initial State

(4,1)

matches Final State

(18,7)

; Initial State

(5,2)

matches Final State

(9,-4)

Each initial state must be tracked through all three stages (

k=1, 2, 3

) using the conditional rule based on whether the preceding value of

X

is even or odd.

Step-by-Step Solution

Evaluate the state transition for Initial State

(2,3)

across stages

k=1, 2, 3

Stage 1:

(8,4)

, Stage 2:

(16,6)

, Stage 3:

(28,9)

Since

X

remains even at every step (

2 \rightarrow 8 \rightarrow 16

Y_k

increases by

k

at each stage.

Evaluate the state transition for Initial State

(3,5)

across stages

k=1, 2, 3

Stage 1:

(13,4)

, Stage 2:

(21,2)

, Stage 3:

(25,-1)

Since

X

remains odd at every step (

3 \rightarrow 13 \rightarrow 21

Y_k

decreases by

k

at each stage.

Evaluate the state transition for Initial State

(4,1)

across stages

k=1, 2, 3

Stage 1:

(6,2)

, Stage 2:

(10,4)

, Stage 3:

(18,7)

Since

X

remains even at every step (

4 \rightarrow 6 \rightarrow 10

Y_k

increases by

k

at each stage.

Evaluate the state transition for Initial State

(5,2)

across stages

k=1, 2, 3

Stage 1:

(9,1)

, Stage 2:

(11,-1)

, Stage 3:

(9,-4)

Since

X

remains odd at every step (

5 \rightarrow 9 \rightarrow 11

Y_k

decreases by

k

at each stage.

Key Concept

Multi-stage recursive state update logic and conditional branch evaluation.

Question

Answer

Step-by-Step Solution

Key Concept

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