1. **Problem Statement:** Find the population of the town in the year 2003 using the Gauss backward difference formula given the population data for years 1980, 1990, 2000, 2010, and 2020. 2. **Given Data:** Years: 1980, 1990, 2000, 2010, 2020 Population (in thousands): 15, 30, 47, 53, 65 3. **Step 1: Define variables and intervals** Let $x_0 = 1980$, and the interval $h = 10$ years (since data is every 10 years). We want to find population at $x = 2003$. Calculate $p = \frac{x - x_n}{h}$ where $x_n = 2020$ (last known year): $$p = \frac{2003 - 2020}{10} = \frac{-17}{10} = -1.7$$ 4. **Step 2: Construct backward difference table** \begin{align*} \text{Year} & : 1980 & 1990 & 2000 & 2010 & 2020 \\ \text{Population} & : 15 & 30 & 47 & 53 & 65 \\ \Delta y & : & & & & \\ \Delta^2 y & : & & & & \\ \Delta^3 y & : & & & & \\ \Delta^4 y & : & & & & \\ \end{align*} Calculate first backward differences $\Delta y$: $$\Delta y_4 = y_5 - y_4 = 65 - 53 = 12$$ $$\Delta y_3 = y_4 - y_3 = 53 - 47 = 6$$ $$\Delta y_2 = y_3 - y_2 = 47 - 30 = 17$$ $$\Delta y_1 = y_2 - y_1 = 30 - 15 = 15$$ Calculate second backward differences $\Delta^2 y$: $$\Delta^2 y_4 = \Delta y_4 - \Delta y_3 = 12 - 6 = 6$$ $$\Delta^2 y_3 = \Delta y_3 - \Delta y_2 = 6 - 17 = -11$$ $$\Delta^2 y_2 = \Delta y_2 - \Delta y_1 = 17 - 15 = 2$$ Calculate third backward differences $\Delta^3 y$: $$\Delta^3 y_4 = \Delta^2 y_4 - \Delta^2 y_3 = 6 - (-11) = 17$$ $$\Delta^3 y_3 = \Delta^2 y_3 - \Delta^2 y_2 = -11 - 2 = -13$$ Calculate fourth backward difference $\Delta^4 y$: $$\Delta^4 y_4 = \Delta^3 y_4 - \Delta^3 y_3 = 17 - (-13) = 30$$ 5. **Step 3: Gauss backward difference formula** The formula for interpolation near the end of the table is: $$ f(x) = y_n + p \Delta y_n + \frac{p(p+1)}{2!} \Delta^2 y_n + \frac{p(p+1)(p+2)}{3!} \Delta^3 y_n + \frac{p(p+1)(p+2)(p+3)}{4!} \Delta^4 y_n $$ Here, $y_n = 65$, $\Delta y_n = 12$, $\Delta^2 y_n = 6$, $\Delta^3 y_n = 17$, $\Delta^4 y_n = 30$. 6. **Step 4: Calculate each term** Calculate factorials: $$2! = 2, \quad 3! = 6, \quad 4! = 24$$ Calculate terms involving $p = -1.7$: $$p(p+1) = (-1.7)(-0.7) = 1.19$$ $$p(p+1)(p+2) = 1.19 \times 0.3 = 0.357$$ $$p(p+1)(p+2)(p+3) = 0.357 \times 1.3 = 0.4641$$ 7. **Step 5: Substitute values into formula** $$ \begin{aligned} f(2003) &= 65 + (-1.7)(12) + \frac{1.19}{2} (6) + \frac{0.357}{6} (17) + \frac{0.4641}{24} (30) \\ &= 65 - 20.4 + 3.57 + 1.0115 + 0.58 \\ &= 65 - 20.4 + 3.57 + 1.0115 + 0.58 \\ &= 49.7615 \end{aligned} $$ 8. **Step 6: Final answer** The estimated population in 2003 is approximately **49.76 thousand**. This completes the interpolation using the Gauss backward difference formula.