1. **Problem Statement:** Find $f(22)$ using the Gauss forward interpolation formula given the data points: $x$: 20, 25, 30, 35, 40, 45 $y$: 354, 332, 291, 260, 231, 204 2. **Understanding the Gauss Forward Formula:** The Gauss forward interpolation formula is used for equally spaced data points and is given by: $$f(x) = y_0 + p\Delta y_0 + \frac{p(p-1)}{2!}\Delta^2 y_0 + \frac{p(p-1)(p-2)}{3!}\Delta^3 y_0 + \cdots$$ where $p = \frac{x - x_0}{h}$, $x_0$ is the first data point, $h$ is the spacing between $x$ values, and $\Delta$ denotes forward differences. 3. **Calculate $h$ and $p$:** $h = 25 - 20 = 5$ $p = \frac{22 - 20}{5} = \frac{2}{5} = 0.4$ 4. **Construct the forward difference table:** | $x$ | $y$ | $\Delta y$ | $\Delta^2 y$ | $\Delta^3 y$ | $\Delta^4 y$ | |-----|-----|------------|--------------|--------------|--------------| | 20 | 354 | -22 | | | | | 25 | 332 | -41 | -19 | | | | 30 | 291 | -31 | 10 | 29 | | | 35 | 260 | -29 | 2 | -8 | -37 | | 40 | 231 | -27 | | | | | 45 | 204 | | | | | Calculations: $\Delta y_0 = 332 - 354 = -22$ $\Delta y_1 = 291 - 332 = -41$ $\Delta y_2 = 260 - 291 = -31$ $\Delta y_3 = 231 - 260 = -29$ $\Delta y_4 = 204 - 231 = -27$ $\Delta^2 y_0 = -41 - (-22) = -19$ $\Delta^2 y_1 = -31 - (-41) = 10$ $\Delta^2 y_2 = -29 - (-31) = 2$ $\Delta^3 y_0 = 10 - (-19) = 29$ $\Delta^3 y_1 = 2 - 10 = -8$ $\Delta^4 y_0 = -8 - 29 = -37$ 5. **Apply the Gauss forward formula:** $$f(22) = y_0 + p\Delta y_0 + \frac{p(p-1)}{2!}\Delta^2 y_0 + \frac{p(p-1)(p-2)}{3!}\Delta^3 y_0 + \frac{p(p-1)(p-2)(p-3)}{4!}\Delta^4 y_0$$ Calculate each term: $p = 0.4$ $p(p-1) = 0.4 \times (0.4 - 1) = 0.4 \times (-0.6) = -0.24$ $p(p-1)(p-2) = -0.24 \times (0.4 - 2) = -0.24 \times (-1.6) = 0.384$ $p(p-1)(p-2)(p-3) = 0.384 \times (0.4 - 3) = 0.384 \times (-2.6) = -0.9984$ Now substitute: $$f(22) = 354 + 0.4 \times (-22) + \frac{-0.24}{2} \times (-19) + \frac{0.384}{6} \times 29 + \frac{-0.9984}{24} \times (-37)$$ Calculate each term: $0.4 \times (-22) = -8.8$ $\frac{-0.24}{2} \times (-19) = -0.12 \times (-19) = 2.28$ $\frac{0.384}{6} \times 29 = 0.064 \times 29 = 1.856$ $\frac{-0.9984}{24} \times (-37) = -0.0416 \times (-37) = 1.5392$ 6. **Sum all terms:** $$f(22) = 354 - 8.8 + 2.28 + 1.856 + 1.5392 = 350.8752$$ 7. **Final answer:** $$\boxed{f(22) \approx 350.88}$$