With a smoothing constant of = 0.2, equation (6.2) shows that the forecast for the 13th week of the gasoline sales data from Table

With a smoothing constant of α = 0.2, equation (6.2) shows that the forecast for the 13th week of the gasoline sales data from Table is given by F13 = 0.2Y12 + 0.8F12. However, the forecast for week 12 is given by F12 = 0.2Y11 + 0.8F11. Thus, we could combine these two results to write the forecast for the 13th week as
F13 = 0.2Y12 + 0.8(0.2Y11 + 0.8F11) = 0.2Y12 + 0.16Y11 + 0.64F11

a. Make use of the fact that F11= 0.2Y10 + 0.8F10 (and similarly for F10 and F9) and continue to expand the expression for F13 until you have written it in terms of the past data values Y12, Y11, Y10, Y9, and Y8, and the forecast for period 8.
b. Refer to the coefficients or weights for the past data values Y12, Y11, Y10, Y9, and Y8; what observation can you make about how exponential smoothing weights past data values in arriving at new forecasts? Compare this weighting pattern with the weighting pattern of the moving averages method.
Equation (6.2) Ft+1 = αYt + (1 - α)Ft

TABLE GASOLINE SALES TIMES SERIES
Week Sales (1000s of gallons)
1 .......... 17
2 .......... 21
3 .......... 19
4 .......... 23
5 ............. 18
6 .......... 16
7 .......... 20
8 .......... 18
9 .......... 22
10 ............ 20
11 ............ 15
12 .......... 22