Estimate 1.02 to the 10th power

Question

Solve in your head. Estimate (1.02)^{10} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Binomial expansion of (1 + x)^n with x = 0.02, n = 10: (1.02)^{10} \approx 1 + 10(0.02) + \binom{10}{2}(0.02)^2 = 1 + 0.2 + 45 	imes 0.0004 = 1.218. The cubic term adds \binom{10}{3}(0.02)^3 = 120 	imes 0.000008 \approx 0.001, giving (1.02)^{10} \approx 1.219. True value: 1.21899\ldots, the two-term-plus-nudge estimate is essentially exact. Takeaway. For compounding estimates, "simple interest plus half the pairwise cross terms": (1+x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2. The second term is the compounding bonus, here 1.8 points on top of the naive 20%. Cross-check via logs: 10 \ln 1.02 …

Estimate 1.02 to the 10th power

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