Estimate 1.01 to the 30th power

Question

Solve in your head. Estimate (1.01)^{30} to three decimal places. Aim: under 20 seconds.

Accepted Answer

Binomial expansion of (1 + x)^n with x = 0.01, n = 30: (1.01)^{30} \approx 1 + 30(0.01) + \binom{30}{2}(0.01)^2 = 1 + 0.3 + 435 	imes 0.0001 = 1 + 0.3 + 0.0435 = 1.3435. The small rate keeps each term modest, but the large exponent makes \binom{30}{2} = 435 sizable, so the compounding bonus is a solid 4.35 points. The cubic term adds \binom{30}{3}(0.01)^3 = 4060 	imes 10^{-6} \approx 0.004, giving (1.01)^{30} \approx 1.348. True value: 1.34785\ldots Takeaway. With a small rate but many periods, the \binom{n}{2}x^2 term carries the compounding: (1+x)^n \approx 1 + nx + \frac{n(n-1)}{2}x^2. H…

Estimate 1.01 to the 30th power

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