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 July 16th, 2012, 02:21 PM #1 Member   Joined: Jun 2012 Posts: 76 Thanks: 0 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.
 July 16th, 2012, 02:39 PM #2 Math Team   Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus Re: mathematical induction question [color=#000000]You want to prove that $1+2+2^2+2^3+\ldots+2^{n-1}=2^{n}-1$. For n=1 you get 1=1 Suppose it holds for n=k, meaning that $1+2+2^{2}+2^{3}+\ldots+2^{k-1}=2^{k}-1$, you will have to prove that the equation holds for n=k+1. Indeed for n=k+1, $\underbrace{1+2+2^2+2^3+\ldots+2^{k-1}}_{=2^{k}-1}+2^{k}=2^{k}-1+2^{k}=2\cdot 2^{k}-1=2^{k+1}-1$.[/color]
 July 16th, 2012, 02:46 PM #3 Member   Joined: Jun 2012 Posts: 76 Thanks: 0 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
 July 16th, 2012, 03:20 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: mathematical induction question Recall the law of exponents: $a^b\cdot a^c=a^{b+c}$ hence: $2\cdot2^k=2^1\cdot2^k=2^{k+1}$
 July 16th, 2012, 07:47 PM #5 Member   Joined: Jun 2012 Posts: 76 Thanks: 0 Re: mathematical induction question ok im lost again. why is it 2x2^k and not 2^k+2^k
 July 16th, 2012, 07:57 PM #6 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: mathematical induction question Both are equivalent: $2^k+2^k=2^k(1+1)=2\cdot2^k=2^{k+1}$
 July 17th, 2012, 10:56 AM #7 Member   Joined: Jun 2012 Posts: 76 Thanks: 0 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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