# How to prove a formula for the sum of powers of 2 by induction?

Here is where I"m getting off traông chồng. Lets look at the right side of the last equation: $2^n+1 -1$ I can rewrite this as the following.

Bạn đang xem: How to prove a formula for the sum of powers of 2 by induction?

$2^1(2^n) - 1$ But, from our hypothesis $2^n = 2^n+1 - 1$ Thus:

$2^1(2^n+1 -1) -1$ This is where I get lost. Because when I distribute through I get this.

$2^n+2 -2 -1$ This is wrong is it not? Am I not applying the rules of exponents correctly here? I have the solution so I know what I"m doing is wrong. Here is the correct proof.

summation induction

Share

Cite

Improve sầu this question

Follow

edited Mar 8 "15 at 4:14

user147263

asked Feb 18 "11 at 0:37

lampShadelampShade

99311 gold badge1212 silver badges1717 bronze badges

$endgroup$

1

Add a bình luận |

## 4 Answers 4

Active Oldest Votes

21

$egingroup$

Both Inductive Step khổng lồ prove sầu is: $ 2^n+1 = 2^n+2 - 1$Our hypothesis is: $2^n = 2^n+1 -1$

are wrong & should be

Inductive sầu Step lớn prove sầu is: $ 2^0 + 2^1 + ... + 2^n + 2^n+1 = 2^n+2 - 1$Our hypothesis is: $ 2^0 + 2^1 + ... + 2^n = 2^n+1-1$Add $2^n+1$ to both sides of the hypothesis and you have the step lớn prove since $2^n+1-1 +2^n+1 = 2^n+2 - 1$

Share

Cite

Improve sầu this answer

Follow

answered Feb 18 "11 at 0:44

HenryHenry

134k99 gold badges108108 silver badges212212 bronze badges

$endgroup$

Add a bình luận |

21

$egingroup$

An easy way to lớn vày this is using binary. Here"s an idea of what I mean:$2^0$ in binary is $1$$2^1$ in binary is $10$$2^2$ in binary is $100$

For a general rule:

$2^n$ in binary is $100cdots0$ (n zeros)

Add those together & you get $2^0 + 2^1 + ... + 2^n$ in binary is $11...11$ ($n+1$ ones).

Now it"s obvious that adding 1 to lớn that gives you$$100cdots00 quad ext ($n+1$ zeros)$$Which as we all know is $2^n+1$.

Thus $2^n+1 - 1$ is equal so the sum of powers of two up lớn $2^n$.

Share

Cite

Follow

edited Jan 11 "16 at 0:30

JnxF

1,23911 gold badge99 silver badges2222 bronze badges

answered Jan 11 "16 at 0:12

AdamAdam

21122 silver badges22 bronze badges

$endgroup$

1

Add a phản hồi |

9

$egingroup$

**HINT**$ $ Here"s the inductive proof for summing a general geometric series.

**THEOREM** $
mquad 1 + x + cdots + x^n-1 = dfracx^n-1x-1$

**Proof** $ $ Base case: It is true for $
m n = 1:,:$ viz. $
m 1 = (x-1)/(x-1):$.

Inductive step: Suppose it is true for $ m n = k:. $ Then we have

$$ m x^k + (x^k-1 +: cdots: + 1): =: x^k +fracx^k-1x-1 = fracx^k+1-1x-1$$

which implies it is true for $
m: n = k+1:,:$ thus completing the inductive sầu proof.

Xem thêm: Bài 20: Thu Hoạch Bảo Quản Và Chế Biến Nông Sản, Thu Hoạch Bảo Quản Và Chế Biến Nông Sản

The proof you seek is just the special case $ m x = 2 $.

Share

Cite

Improve this answer

Follow

answered Feb 18 "11 at 1:19

Bill DubuqueBill Dubuque

250k3333 gold badges249249 silver badges799799 bronze badges

$endgroup$

Add a bình luận |

0

$egingroup$

I don"t see the answer I like here, so I"m writing my own.

Basic proof:

We wish to prove $2^0 + 2^1 + ... + 2^n-1 = 2^n - 1$ for all $n$. We can verify by inspection this is true for n=1. Next, assume that $2^0 + 2^1 + ... + 2^n = 2^n+1 - 1$.

$(2^0 + 2^1 + ... + 2^n) + 2^n+1 = (2^n+1 - 1) + 2^n+1 = 2 cdot 2^n+1 - 1 = 2^n+2 - 1$, so we have shown $2^0 + 2^1 + ... + 2^n-1 = 2^n - 1$ is true for all n.

Share

Cite

Follow

edited Jan 18 at 7:52

Community♦

1

answered Atruyền thông quảng cáo 9 "18 at 4:12

EratosthenesEratosthenes

10122 bronze badges

$endgroup$

2

Add a phản hồi |

## Your Answer

Thanks for contributing an answer to pgdtxhoangmai.edu.vnematics Staông chồng Exchange!

Please be sure to lớn*answer the question*. Provide details & chia sẻ your research!

But *avoid* …

Use pgdtxhoangmai.edu.vnJax to format equations. pgdtxhoangmai.edu.vnJax reference.

To learn more, see our tips on writing great answers.

Xem thêm: Giải Bài Tập Sinh Học 8 Bài 38 : Bài Tiết Và Cấu Tạo Hệ Bài Tiết Nước Tiểu

Draft saved

Draft discarded

### Sign up or log in

Sign up using Google

Sign up using Facebook

Sign up using Thư điện tử & Password

Submit

### Post as a guest

Name

Email Required, but never shown

### Post as a guest

Name

Required, but never shown

Post Your Answer Disthẻ

By clicking “Post Your Answer”, you agree khổng lồ our terms of service, privacy policy & cookie policy

## Not the answer you're looking for? Browse other questions tagged summation induction or ask your own question.

Featured on Meta

16 votes · comment · stats

Linked

0

How to lớn prove sầu by induction that $sum^n_i=12^i-1=2^n-1$?

1

Proof by induction (exponents)

4

Find all positive sầu integers $n$ such that $frac2^n-1+1n$ is integer. Where I'm wrong?

1

Proof By Induction Help?

0

No disjoint partition for $2^1, 2^2, …,2^15$ such the sum of elements in all partitions is the same

1

Proof by induction that $sum_j=0^n 2^j = 2^n+1 - 1$

0

By induction, show that for ∀n∈N, it is true that:

Related

2

Prove by induction (formula for $sum^n(3i-1)^2$)

0

Prove by induction $ sum^n_i=1(i-1/2) = n^2/2 $

0

Trying to correctly write the proof using *strong* induction of the sum of the nth positive integer

1

Proof of closed khung for finite sums by induction.

0

Proof by Induction of an ineunique with a sum

1

Am I properly using induction (specifically the induction hypothesis)?

1

Inductive sầu proof on a sequence

0

Induction on powers of powers.

1

How bởi vì I prove that the phối of rational functions is closed under composition by structural induction?

Hot Network Questions more hot questions

Question feed

Subscribe khổng lồ RSS

Question feed To subscribe khổng lồ this RSS feed, copy và paste this URL inlớn your RSS reader.

pgdtxhoangmai.edu.vnematics

Company

Stack Exchange Network

site design / biểu tượng logo © 2021 Staông xã Exchange Inc; user contributions licensed under cc by-sa. rev2021.7.8.39684

pgdtxhoangmai.edu.vnematics Stachồng Exchange works best with JavaScript enabled

Your privacy

By clicking “Accept all cookies”, you agree Staông chồng Exchange can store cookies on your device và discđại bại information in accordance with our Cookie Policy.