SOLUTION: Hello this is a mathematical induction prove question I need help with. 1. Show that, for every positive integer n: 1 + 2 + 2^2 + 2^3 + … + 2^(n-1) = 2^n - 1

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Question 1183173: Hello this is a mathematical induction prove question I need help with.

1. Show that, for every positive integer n:
1 + 2 + 2^2 + 2^3 + … + 2^(n-1) = 2^n - 1
Found 2 solutions by Edwin McCravy, robertb:
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!



We show that it's true for n=1.



Then we find out what would happen if there were some positive integer k,
for which it were true.  If it were true for n = k, then we would have:



If that were true, we would be able to add 2k to both sides and
it would still be true.  Then we would have:









That is the original expression with k+1 substituted for n.

So if we could find a positive integer k so that the equation were true,
then we would know that it would have to also be true for the next higher
positive integer.

Now we DO have a positive integer k=1 so that the equation is true! That's
because at the first we showed that it is true when n=k=1, so that means it
is true for n=2.  Now, the fact that it's true for n=2 shows that it's also
true for n=3, and so on and on, through all the positive integers.

Edwin
Answer by robertb(5830)   (Show Source): You can put this solution on YOUR website!
This is an alternative solution.
It is easy to see by direct multiplication that

(1-x)*(1 + x + x^2 + x^3 +...+ x^(n-1)) = 1 - x^n.

If you let x = 2 in the above equation, you will get

(-1)*(1 + 2 + 2^2 + 2^3 +...+ 2^(n-1)) = 1-2^n

This implies then that

1 + 2 + 2^2 + 2^3 +...+ 2^(n-1) = 2^n -1,

after multiplying both sides of the equation by -1.



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