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.
1 answer