Question

plz help me with it

Fixing a ER-{0}, consider the recursive functions f,g:Z+ + R defined as follows: f(1) = g(1) = a while f(n+1) = af(n) and g(n + 1) = g(n)". (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 ad- dition and multiplication enjoy, that is illustrated by the difference between f and g? As a remark, the function is called tetration.

Answer 1

Given, bee) = g(L) =a .. front]) = a An) .....Ù g(n+1) = g(n)a .... Put, n=lno, 6(2) = aber) a n=2, 6(3) = a (2) = 0 n=3, Bla): afte) Put, n=1, in ui 9(2) = 369-aa in 903) = 902)2 = (aqje n = 2 qe? n=3, 3(4) = g(g) = (222) a = gaie - ga3 n = 4, 9 (5) = 3 (4) a = (a 23/9 = Baste Put, n = (2-1), g(n) = a a 10 10 10 when a = 10 P13) g (3) = 10/02 - 10000 16) gin) - aa 19 Exponential lack associative property.

Consider the following example, 49 - 256 (22)9)* = () 216 + 256 In 44, it we peut a=2, we see that R4) = 2 + ((?)). so, associative property lacks in 64). But, in g (A) = 223 = 28 - 256 This is the difference between 8m) and gen). Associative property addition and multiplication enjoy. H2) (803). M4)= QQ ( 22 qala) (442). 63)) •424) = (anal). = a arataaa som bl2) (H3). H4)) = (R ().63)). H4). Associative property of multiplication holds in bln). 909) (909.9(4)) = 29.(a aa) = a(aa'+a3) qQ+la?+ 3) q&)423 (969).909).g(4) = 109. aa?).. a = (a +91).q23 (.' Associative Paphy old 4 a94a3 in ) Associative property of multiplication holds in a gin.