M2 Mathematical Induction

(a)Assume that P(k) is true for some positive integers,
e.g. 1+2+2^2+2^3+...+2^(k-1)=2^k-1 where k is a positive integer
When n=k+1
LHS= (1+2+2^2+2^3+...+2^(k-1)) +2^k
= 2^k-1+2^k
= 2*2^k-1
= 2^(k+1)-1
= RHS
Therefore, P(k+1) is true.


(b)Let P(n) be the statement '1+2+2^2+2^3+...+2^(n-1)=2^n-1 ' for all positive integer n

When n =1
LHS=1
RHS=2^1-1 =1
AS LHS=RHS
Therefore P(1) is true
照抄（a）
By MI, P(n) is true for all positive integers n.





其實個問題1條過仲好。不知道你這問題的marking scheme是怎麼的。