A subscription service evaluates customer activity over a three-month period (k = 1, 2, 3)… — GMAT

A subscription service evaluates customer activity over a three-month period ( $k = 1, 2, 3$ ). At the end of each month $k$ , a customer's total loyalty points $P_k$ update based on their points from the previous month $P_{k-1}$ and the net points earned during month $k$ ( $\Delta_k$ ), according to the state transition rule:

P_k = \lfloor 0.8 \times P_{k-1} \rfloor + \Delta_k

where $\lfloor x \rfloor$ represents the greatest integer less than or equal to $x$ .

At the end of Month 3 ( $k = 3$ ), membership tiers are assigned based on final points $P_3$ :
- VIP Tier: $P_3 \ge 150$
- Premium Tier: $80 \le P_3 < 150$
- Standard Tier: $P_3 < 80$

Match each initial customer profile (defined by initial points $P_0$ and monthly additions $\Delta_1, \Delta_2, \Delta_3$ ) to its corresponding final state at the end of Month 3.

Profile 1: $P_0 = 100$ ; additions $\Delta_1 = 50, \Delta_2 = 10, \Delta_3 = 40$ Final Points: 131 (Premium Tier)
Profile 2: $P_0 = 150$ ; additions $\Delta_1 = 30, \Delta_2 = 40, \Delta_3 = 50$ Final Points: 178 (VIP Tier)
Profile 3: $P_0 = 90$ ; additions $\Delta_1 = 0, \Delta_2 = 10, \Delta_3 = 20$ Final Points: 73 (Standard Tier)
Profile 4: $P_0 = 200$ ; additions $\Delta_1 = 0, \Delta_2 = 10, \Delta_3 = 30$ Final Points: 140 (Premium Tier)

Answer

Profile 1 matches Final Points: 131 (Premium Tier); Profile 2 matches Final Points: 178 (VIP Tier); Profile 3 matches Final Points: 73 (Standard Tier); Profile 4 matches Final Points: 140 (Premium Tier).

Each profile is evaluated by computing the exact state update

P_k = \lfloor 0.8 \cdot P_{k-1} \rfloor + \Delta_k

sequentially over three transitions. Profile 1 yields 131 points (Premium Tier), Profile 2 yields 178 points (VIP Tier), Profile 3 yields 73 points (Standard Tier), and Profile 4 yields 140 points (Premium Tier).

Step-by-Step Solution

Apply the state transition formula

P_k = \lfloor 0.8 \times P_{k-1} \rfloor + \Delta_k

iteratively across months

k = 1, 2, 3

for Profile 1.

P_1 = 80 + 50 = 130

;

P_2 = 104 + 10 = 114

;

P_3 = 91 + 40 = 131

. Tier: Premium.

Tracking sequential updates ensures each intermediate decay and point addition step is accounted for accurately.

Apply the state transition formula iteratively across months

k = 1, 2, 3

for Profile 2.

P_1 = 120 + 30 = 150

;

P_2 = 120 + 40 = 160

;

P_3 = 128 + 50 = 178

. Tier: VIP.

Continuous accumulation and decay calculation determines the final state threshold reached.

Apply the state transition formula iteratively across months

k = 1, 2, 3

for Profile 3.

P_1 = 72 + 0 = 72

;

P_2 = 57 + 10 = 67

;

P_3 = 53 + 20 = 73

. Tier: Standard.

Floor function truncation must be evaluated at each step prior to adding monthly points.

Apply the state transition formula iteratively across months

k = 1, 2, 3

for Profile 4.

P_1 = 160 + 0 = 160

;

P_2 = 128 + 10 = 138

;

P_3 = 110 + 30 = 140

. Tier: Premium.

Evaluating high starting values shows how retention decay influences final tier classification.

Key Concept

Sequential state transitions with floor function decay and additive step updates

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