Solve the difference equation x_{t+1}=0.99x_t-4, t=0,1,2,\ldots, with x_0=100. What is the value of x_{66}? Round your answer to two decimal places.

Question

BlackTom AI · Accepted Answer

Step 1: Restate the problem in my own words to ensure understanding. We have a first-order difference equation x_{t+1} = 0.99 x_t - 4 with initial condition x_0 = 100, and we need x_{66}.

Step 2: Find the fixed point (steady-state) of the recurrence. If x_{t+1} = x_t = x*, then x* = 0.99 x* - 4. Solving gives x* - 0.99 x* = -4, i.e., 0.01 x* = -4, so x* = -400.

Step 3: Write the general solution. For a linear difference equation of the form x_{t+1} = a x_t + b, the solution is x_t = x* + (x_0 - x*) a^t, where a = 0.99 and x* = -400. Therefore:
 x_t = -400 + (100 - (-400)) (0.99)^t = -400 + 500 (0.99)^t.

Step 4: Evaluate at t = 66. Compute x_{66} = -400 + 500 (0.99)^{66}.

Step 5: Approximate (0.99)^{66}. A common approach is to use natural logarithms: (0.99)^{66} = exp(66 * ln(0.99)). Since ln(0.99) ≈ -0.01005033585, we get 66 * ln(0.99) ≈ -0.663322, and exp(-0.663322) ≈ 0.515. A slightly more careful approximation yields a value around 0.51 to 0.515.

Step 6: Plug this into the expression. If (0.99)^{66} ≈ 0.515, then x_{66} ≈ -400 + 500 * 0.515 ≈ -400 + 257.5 ≈ -142.5. If (0.99)^{66} ≈ 0.511, then x_{66} ≈ -400 + 500 * 0.511 ≈ -400 + 255.5 ≈ -144.5. In any reasonable precision, x_{66} is around -142 to -145, i.e., approximately -143 to -144 when rounded to two decimals.

Step 7: Compare with the provided value -379.54. To understand where that could come from, notice that the expression x_t = -400 + 500 (0.99)^t cannot yield as low as -379.54 unless (0.99)^66 were about 0.0789 (since -400 + 500*0.0789 ≈ -400 + 39.45 ≈ -360.55, which is still not -379.54). This indicates a miscalculation or a misinterpretation of the recurrence parameters or index. The correct method and calculation show x_{66} is far from -379.54.

Step 8: Final assessment. The exact closed form is x_{66} = -400 + 500 (0.99)^{66}, which numerically is about -142.5 to -144.5, depending on the precise value of (0.99)^{66} used in the approximation. The given answer -379.54 does not align with the derived formula or the typical numeric evaluation of (0.99)^{66}.

Solve the difference equation x_{t+1}=0.99x_t-4, t=0,1,2,\ldots, with x_0=100. What is the value of x_{66}? Round your answer to two decimal places.数值题

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