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I'm reading the book "Singularity is near", and there is a passage where the author says: "It takes 100 years to achieve this, with current rate of progress, but because we're doubling the rate of progress every decade, we'll achieve a progress of century in 25 years".

Can you please help me with the calculation,i.e. the 25 years?

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up vote 1 down vote accepted

I want to eat 100 Mars bars. At my current rate of 1 Mars bar a year, it will take me 100 years. But if I start eating 2 a year after 10 years, and 4 a year after 20 years, then in 25 years I'll eat $$10\times1+10\times2+5\times4=50$$ Seems to me it would actually take 32 years: $$10\times1+10\times2+10\times4+4\times8=102$$ But this is assuming the increase is discrete, happening only at the end of each 10-year period. Most likely, the author has a continuous model in mind, and one needs to perform an integration rather than an addition. How are you on integral calculus?

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thanks for your ans. I should be pretty comfortable with integral and calculus. – Pep Jul 20 '13 at 6:42
OK, then; I eat at the rate $f(t)=e^{(t\log2)/10}$ Mars bars a year at time $t$, measuring $t$ in years. To begin with, $t=0$, 1 Mars bar a year, 100 years. But over 25 years, $\int_0^{25}f(t)\,dt$. – Gerry Myerson Jul 20 '13 at 6:52
hmm, would it be possible for you to work the other way around, i.e. derive 25 years, given the other informations? – Pep Jul 20 '13 at 7:01
Solve $100=\int_0^Tf(t)\,dt$ for $T$. I haven't worked it out, and I don't guarantee you'll get 25 --- the author may have been doing some rounding to the nearest nice number. – Gerry Myerson Jul 20 '13 at 7:30

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