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July 16th, 2012, 02:21 PM   #1
ehh
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mathematical induction question

this is what i have

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

step 1: Prove n=1
2^1-1 = 1

step 2: assume true for n=k
1+2+2^2+....+2^(k-1) = 2^(k) -1

step 3: Find next term n=k+1
1+2+2^2+....+2^(k-1) + 2^((k+1)-1) = 2^(k+1) -1

step 4: show true for n=k+1
1+2+2^2+....+2^(k-1) = 2^(k) -1
1+2+2^2+....+2^(k-1) + 2^((k+1)-1) = 2^(k) -1 + 2^((k+1)-1) = 2^k -1 +2^k = ? this is where i am stuck.

i know the answer is 2^(k+1)-1 but i dont know how to get there from where i am. please help.
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July 16th, 2012, 02:39 PM   #2
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Re: mathematical induction question

[color=#000000]You want to prove that .

For n=1 you get 1=1

Suppose it holds for n=k, meaning that , you will have to prove that the equation holds for n=k+1.

Indeed for n=k+1,

.[/color]
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July 16th, 2012, 02:46 PM   #3
ehh
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Re: mathematical induction question

ahhh ok i didnt know that 2x2^k = 2^(k+1)

that is where i was getting stuck. thanks! huge help
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July 16th, 2012, 03:20 PM   #4
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Re: mathematical induction question

Recall the law of exponents:



hence:

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July 16th, 2012, 07:47 PM   #5
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Re: mathematical induction question

ok im lost again. why is it 2x2^k and not 2^k+2^k
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July 16th, 2012, 07:57 PM   #6
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Re: mathematical induction question

Both are equivalent:

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July 17th, 2012, 10:56 AM   #7
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Re: mathematical induction question

ok i get it. i asked the teacher about it as well and she explained it some more.
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