QuestionQuestion

Tetration Fixing a € R -{0}, consider the recursive functions f,g : Z+ ->R defined as follows: f(1) = g(1) = a while f (n + 1) = aᶠ(ᶯ) and g(n + 1) = g(n)ᵃ
(a) Write out expressions for f(3) and g(3) in terms of a, and compute their values in case a = 10.
(b) Give a simplified expression for g(n), for an arbitrary n.
(c) What is the property that exponentiation lacks, a property that addition and multiplication enjoy, that is illustrated by the difference between f and g? As a remark, the function is called tetration.

Explain in your own words, making up a metaphor if you like, why induction is a valid method of proof of ɏnP(n), where P is some predicate on N.

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Induction is a valid method of proof because this method checks the truthness of the statement at all values of n. Also, each truth value of the statement is based on the previous statement. For example for n =1, P(1) -->...

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