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Topic:  Mathematical Induction 
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Subject:  RE: Mathematical Induction 
Author:  Laura 
Date:  Nov 11 2003 
E(2^(i1))=2^(n)1.
Now, the E sigma sign means that i takes every value from 1 to n like this:
E(2^(i1))=2^(11)+2^(21)+2^(31)+...+2^(n1)=2^0+2^1+2^2+...+2^(n1).
Now, from what you said, they have given you the answer, that is that the sum
equals 2^(n)1, and you have to prove that it is corect. There are some ways
to do this, but by mathematical induction you do it like this:
There are 2 steps. The first one is to check if the relatioship is correct for
n=1 and n=2. That is check if:
E2^(i1)=2^(n)1 for n=1 and n=2.
that means: 2^0=2^11 and 2^0+2^1=2^21.(You have to do the math here)
The second step: you assume that the relationship is correct for n, and you
prove it for the value of n+1.
So:
You assume that E2^(i1){i from 1 to n}=2^n1
And you have to prove that
E2^(i1){i from 1 to n+1}=2^(n+1)1
You do thus by writing
E2^(i1){i from 1 to n+1}=E2^(i1){i from 1 to
n}+2^(n+11)=2^(n)1+2^n=2^n+2^n1=2^n(1+1)1=2^(n)*21=2^(n+1)1.
Thus, what you wanted to prove is proven.
For any given n, the relation works for n+1. (If it works for n=1, it will work
for n=2 the for n=3 and so on).
Note that this is for this specific example. Any other one works the same(eg
E(i)=i(i+1)/2). The steps are exactly the same(check for n=1 and n=2, assume
it's corect for n and check for n+1), the only thing that changes is the
math.
 
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