Let's break this down step-by-step:

1. Three-Month Moving Average: We will calculate the average of the past three months for each month, beginning from the third month. The data provided is:
   $240, 350, 230, 260, 280, 320, 220, 310, 240, 310, 240, 230$

   To find the three-month moving averages:

   • For month 3: $\frac{240 + 350 + 230}{3} = \frac{820}{3} \approx 273.33$
   • For month 4: $\frac{350 + 230 + 260}{3} = \frac{840}{3} = 280$
   • For month 5: $\frac{230 + 260 + 280}{3} = \frac{770}{3} \approx 256.67$
   • For month 6: $\frac{260 + 280 + 320}{3} = \frac{860}{3} \approx 286.67$
   • For month 7: $\frac{280 + 320 + 220}{3} = \frac{820}{3} \approx 273.33$
   • For month 8: $\frac{320 + 220 + 310}{3} = \frac{850}{3} \approx 283.33$
   • For month 9: $\frac{220 + 310 + 240}{3} = \frac{770}{3} \approx 256.67$
   • For month 10: $\frac{310 + 240 + 310}{3} = \frac{860}{3} \approx 286.67$
   • For month 11: $\frac{240 + 310 + 240}{3} = \frac{790}{3} \approx 263.33$

   The three-month moving averages are:
   $273.33, 280, 256.67, 286.67, 273.33, 283.33, 256.67, 286.67, 263.33$

2. Exponential Smoothing: Using $\alpha = 0.2$, we need to apply exponential smoothing to forecast future values. The formula is: $S_t = \alpha \cdot X_t + (1 - \alpha) \cdot S_{t-1}$ where $S_t$ is the smoothed value at time $t$, and $X_t$ is the actual value at time $t$. We will take the first observation as the initial smoothed value. Let's start:

   • $S_1 = 240$ (initial value)
   • $S_2 = 0.2 \times 350 + 0.8 \times 240 = 270$
   • $S_3 = 0.2 \times 230 + 0.8 \times 270 = 262$
   • $S_4 = 0.2 \times 260 + 0.8 \times 262 = 261.6$
   • $S_5 = 0.2 \times 280 + 0.8 \times 261.6 = 265.28$
   • (S_6 = 0.2 \times 320 +