Posted by **Jane** on Friday, November 23, 2007 at 3:32pm.

I have solved parts of the question, but I want to know if they are correct. Other parts I'm not sure how to do, like part b and c.

g(x)=2x+1

Define g^3= g^2 o g,...,g^n =g^(n-1) o g

(where "o" means composition)

a) Give rules for g^2, g^3, g^4

Ans: g^2= g o g= g(2x+1)= 4x+3

g^3= g^2 o g= g^2(2X+1)=8X+7

g^4= g^3 o g= g^3(2x+1)= 16x+15

b) Make a conjecture as to the general rule for g^n. for any positive integer n.

Ans: ? g^n= 2^(n-1) 2x+ 2^(n) -1

c) Verify conjecture by induction.

Ans:

p(n):g^n

p(n)==> p(n+1)

Assume p(n),

prove g^(n+1)= 2^(n) 2x+ 2^(n+1)-1

*** (I'm not if the following is correct.)***

g^(n+1)=2^(n-1)2x+2^(n)-1 +(g^(n) o g)

If correct, how do I continue.

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