Estimate 0.99 to the 20th power

Question

Something loses 1\% of its value each period for 20 periods, so the surviving fraction is (0.99)^{20}. Estimate (0.99)^{20} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Binomial expansion of (1 - x)^n with x = 0.01, n = 20. The signs alternate: (0.99)^{20} \approx 1 - 20(0.01) + \binom{20}{2}(0.01)^2 = 1 - 0.2 + 190 	imes 0.0001 = 1 - 0.2 + 0.019 = 0.819. So the naive -20\% overshoots the loss; the quadratic term adds back 0.019. The cubic term subtracts a whisker, \binom{20}{3}(0.01)^3 = 1140 	imes 10^{-6} \approx 0.001, giving (0.99)^{20} \approx 0.818. True value: 0.81791\ldots, spot on. Takeaway. Decay compounds gentler than linear intuition suggests: (1 - x)^n \approx 1 - nx + \frac{n(n-1)}{2}x^2, and the positive quadratic term means you keep more th…

Estimate 0.99 to the 20th power

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