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I understand than function decreases exponentially, then order of growth of this function is exponential with negative exponent. But what does it mean that order of growth decreases exponentially?

I don't understand what should i look for in this exercise:

Exercise 2.33 Find a recurrence describing a sequence for which the order of growth decreases exponentially for odd-numbered terms, but increases exponentially for even-numbered terms.

Could you provide several first items of such a sequence to let me understand the exercise?

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I think it just wants a recurrence for a sequence where the even-numbered terms increase exponentially and the odd-numbered terms decrease exponentially. Talking about the order of growth changing sounds like some kind of second-derivative property but I think it’s just poor phrasing.

To be absolutely sure, you’d have to ask the person who wrote the exercise.

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    $\begingroup$ So, 32,1, 16,2, 8,4, 4,8, 2,16, 1,32,... or 1,-1, 2,-2, 4,-4, 8,-8, ...? So, is it like $a^x$ and $a^{-x}$ or $a^x$ and $-a^x$. $\endgroup$ – Yola May 18 '18 at 9:47
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    $\begingroup$ Ha! I hadn’t even thought of that ambiguity! Either one could be correct. I suggest you give both answers: nothing makes somebody fix their exercise faster than an answer that says, “I don’t know what you mean. If you mean X, then the answer is A. If you mean Y, then the answer is B.” $\endgroup$ – David Richerby May 18 '18 at 9:50

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