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Can someone explain this prove mathematical induction question?

Use mathematical induction to prove the following expression: $$ \sum_{k = 1}^{n} 2^{n + 1} - 1. $$ I tried my best to solve it but when i tried to prove for p(1) it got failed and for p(0) it is working fine.

I am new to mathematical induction and have test tomorrow.

I am new to this maths stack exchange and please for give if there is any formatting mistakes.

Tried to solve this

Tried to solve this - Continued

Please help me out and thanks.

Community
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Himanshu Chawla
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  • Note that $\sum_{i=1}^n a_i= a_n+\sum_{i=1}^{n-1} a_i$ for $n\ge1$ by the only way to rigorously define $\sum$ – Hagen von Eitzen Nov 11 '18 at 13:52
  • Maybe your problem is related to the fact that the given identity is wrong. Try with the correct version that is $\sum_{i=0}^n2^i=2^{n+1}-1$. – user Nov 11 '18 at 13:55

1 Answers1

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There is a typo in the text, it should be

$$\sum_{i=0}^n2^i=2^{n+1}-1$$

then proceed firstly by base case and then by induction step assuming true $P(n)$ and deriving from it $P(n+1)$.

Refer also to the related

user
  • 154,566
  • or it should be $\ldots =2^{n+1}-2$ – Hagen von Eitzen Nov 11 '18 at 13:52
  • @HagenvonEitzen Equivalentelly yes of course, but referring to the usual expression for the geometric series I think that the typo is likely to be that for $i=0$. – user Nov 11 '18 at 13:54
  • @gimusi Thanks for the answering the query. Can you please also clear the solution once ? I know i should do it myself but it has become very confusing now. Can you ? – Himanshu Chawla Nov 11 '18 at 13:59
  • @HimanshuChawla No I can't but I can give you a further tip. Start from $$\sum_{i=0}^{n+1}2^i=2^{n+1}+\sum_{i=0}^n2^i=\ldots$$ and try to show that $$\sum_{i=0}^{n+1}2^i=2^{n+2}-1$$ Also take a look to others induction proof you can find here on MSE and practice a lot on that. – user Nov 11 '18 at 14:03
  • @HimanshuChawla Refer for example to OP – user Nov 11 '18 at 14:08
  • @gimusi I have just tried to solve the question on my own and got stucked at the last step. I hope now you can help me out. Attaching the images – Himanshu Chawla Nov 11 '18 at 14:49
  • Try to follow the hint I gave here two comments above. I’ll take a look to your work. To do that you should be confident with induction method. If you have doubts on that you could start proving simpler identity as for example $\sum_{i=1}^n i=n(n+1)/2$. Have you already proved that one? – user Nov 11 '18 at 14:59
  • @gimusi I have just edited and added one more step to my answer. Please check it once and just give me the hint or correct me up where i am wrong. I know i am at the very last step to get it proved. – Himanshu Chawla Nov 11 '18 at 15:08
  • The base case is ok and it suffices to prove P(1), I don’t understand what you are doing for the induction step, we need to consider the sum as shown here above. – user Nov 11 '18 at 15:19