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Difficulty: MediumSequential Processes and State Transitions

A digital subscription platform updates user account balances for Reward Points (PP) and Bonus Credits (CC) at the end of each month kk (where k=1,2,3k = 1, 2, 3) according to the following state transition rules:

- Pk=0.8Pk−1+50P_k = 0.8 P_{k-1} + 50
- Ck=Ck−1+0.1Pk−1C_k = C_{k-1} + 0.1 P_{k-1}

At month k=0k = 0, an account starts with P0=500P_0 = 500 points and C0=100C_0 = 100 credits.

Which of the following statements regarding the account balances are true? Select all that apply.

  1. The promotional credit balance at the end of Month 2 (C2C_2) is equal to 195.Answer
  2. The tier status point balance at the end of Month 3 (P3P_3) is equal to 378.Answer
  3. C
    The promotional credit balance at the end of Month 3 (C3C_3) is equal to 241.
  4. D
    The tier status point balance at the end of Month 2 (P2P_2) is equal to 450.
  5. E
    The tier status point balance decreases by exactly 50 points from Month 1 to Month 2.

Answer

The statement that the promotional credit balance at the end of Month 2 is 195 and the statement that the tier status point balance at the end of Month 3 is 378 are both correct.
Evaluating the sequential transitions stage-by-stage gives P1=450P_1 = 450, C1=150C_1 = 150, P2=410P_2 = 410, C2=195C_2 = 195, P3=378P_3 = 378, and C3=236C_3 = 236. Thus, the statement asserting C2=195C_2 = 195 is correct, and the statement asserting P3=378P_3 = 378 is correct.

Step-by-Step Solution

1
Calculate state values at Month 1 (k=1k = 1)
P1=0.8(500)+50=450P_1 = 0.8(500) + 50 = 450 points; C1=100+0.1(500)=150C_1 = 100 + 0.1(500) = 150 credits.
Apply state transition rules using initial values P0=500P_0 = 500 and C0=100C_0 = 100.
2
Calculate state values at Month 2 (k=2k = 2)
P2=0.8(450)+50=410P_2 = 0.8(450) + 50 = 410 points; C2=150+0.1(450)=195C_2 = 150 + 0.1(450) = 195 credits.
Apply state transition rules using Month 1 values (P1=450P_1 = 450, C1=150C_1 = 150).
3
Calculate state values at Month 3 (k=3k = 3)
P3=0.8(410)+50=378P_3 = 0.8(410) + 50 = 378 points; C3=195+0.1(410)=236C_3 = 195 + 0.1(410) = 236 credits.
Apply state transition rules using Month 2 values (P2=410P_2 = 410, C2=195C_2 = 195).
4
Evaluate candidate statements
C2=195C_2 = 195 is true; P3=378P_3 = 378 is true; C3=236≠241C_3 = 236 \neq 241 is false; P2=410≠450P_2 = 410 \neq 450 is false; change in points P1−P2=450−410=40≠50P_1 - P_2 = 450 - 410 = 40 \neq 50 is false.
Compare calculated multi-stage process values directly against each statement.

Key Concept

Sequential Processes and State Transitions
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Topics

Sequential Processes and State Transitions

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