How do you find the sum of square numbers?
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 do you find the sum of square numbers?
$2^1(2^n) - 1$ 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 sầu the solution so I know what I"m doing is wrong. Here is the correct proof.
summation induction
Share
Cite
Improve this question
Follow
edited Mar 8 "15 at 4:14
user147263
asked Feb 18 "11 at 0:37
lampShadelampShade
99311 gold badge1111 silver badges1717 bronze badges
$endgroup$
1
Add a phản hồi |
4 Answers 4
Active sầu Oldest Votes
21
$egingroup$
Both Inductive Step to prove is: $ 2^n+1 = 2^n+2 - 1$Our hypothesis is: $2^n = 2^n+1 -1$
are wrong & should be
Inductive sầu Step to 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 $2^n+1$ khổng lồ both sides of the hypothesis and you have sầu the step to 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
132k88 gold badges107107 silver badges210210 bronze badges
$endgroup$
Add a bình luận |
21
$egingroup$
An easy way lớn do 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 and you get $2^0 + 2^1 + ... + 2^n$ in binary is $11...11$ ($n+1$ ones).
Now it"s obvious that adding 1 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 khổng lồ $2^n$.
Share
Cite
Follow
edited Jan 11 "16 at 0:30
JnxF
1,23811 gold badge99 silver badges2222 bronze badges
answered Jan 11 "16 at 0:12
AdamAdam
21122 silver badges22 bronze badges
$endgroup$
1
Add a comment |
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 sầu 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: Ví Dụ Về Hệ Sinh Thái Là Gì? Phân Loại Hệ Sinh Thái Trong Đời Sống
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
248k3333 gold badges247247 silver badges792792 bronze badges
$endgroup$
Add a bình luận |
0
$egingroup$
I don"t see the answer I lượt thích here, so I"m writing my own.
Basic proof:
We wish lớn prove sầu $2^0 + 2^1 + ... + 2^n-1 = 2^n - 1$ 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 Aquảng bá 9 "18 at 4:12
EratosthenesEratosthenes
10122 bronze badges
$endgroup$
2
Add a comment |
Your Answer
Thanks for contributing an answer lớn hibs.vnematics Staông xã Exchange!
Please be sure to lớn answer the question. Provide details and tóm tắt your research!But avoid …
Asking for help, clarification, or responding to other answers.Making statements based on opinion; bachồng them up with references or personal experience.Use hibs.vnJax to format equations. hibs.vnJax reference.
To learn more, see our tips on writing great answers.
Draft saved
Draft discarded
Sign up or log in
Sign up using Google
Sign up using Facebook
Sign up using E-Mail & Password
Submit
Post as a guest
Name
Email Required, but never shown
Post as a guest
Name
Thư điện tử
Required, but never shown
Post Your Answer Disthẻ
By clicking “Post Your Answer”, you agree to lớn our terms of service, privacy policy và cookie policy
Not the answer you're looking for? Browse other questions tagged summation induction or ask your own question.
Featured on Meta
26 votes · bình luận · stats
Linked
0
How khổng lồ prove 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 lớn correctly write the proof using *strong* induction of the sum of the nth positive integer
1
Proof of closed form 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 proof on a sequence
0
Induction on powers of powers.
1
How vì chưng I prove sầu that the mix of rational functions is closed under composition by structural induction?
Hot Network Questions more hot questions
Question feed
Subscribe to lớn RSS
Question feed To subscribe khổng lồ this RSS feed, copy & paste this URL inkhổng lồ your RSS reader.
hibs.vnematics
Company
Staông chồng Exchange Network
site thiết kế / biểu tượng logo © 2021 Staông chồng Exchange Inc; user contributions licensed under cc by-sa. rev2021.5.28.39383
hibs.vnematics Staông xã Exchange works best with JavaScript enabled
Your privacy
By clicking “Accept all cookies”, you agree Staông xã Exchange can store cookies on your device and discthua thảm information in accordance with our Cookie Policy.
