How to prove a sum formula about powers of 2 and 3? The Next CEO of Stack OverflowProve:...

How to get from Geneva Airport to Metabief, Doubs, France by public transport?

Unclear about dynamic binding

How many extra stops do monopods offer for tele photographs?

Is the D&D universe the same as the Forgotten Realms universe?

Prepend last line of stdin to entire stdin

When you upcast Blindness/Deafness, do all targets suffer the same effect?

What happened in Rome, when the western empire "fell"?

What was the first Unix version to run on a microcomputer?

Domestic-to-international connection at Orlando (MCO)

Does Germany produce more waste than the US?

0 rank tensor vs 1D vector

Is there always a complete, orthogonal set of unitary matrices?

How to edit “Name” property in GCI output?

WOW air has ceased operation, can I get my tickets refunded?

Why, when going from special to general relativity, do we just replace partial derivatives with covariant derivatives?

Bartok - Syncopation (1): Meaning of notes in between Grand Staff

Does increasing your ability score affect your main stat?

Why don't programming languages automatically manage the synchronous/asynchronous problem?

Is it my responsibility to learn a new technology in my own time my employer wants to implement?

Is there a way to save my career from absolute disaster?

Is there a difference between "Fahrstuhl" and "Aufzug"

Method for adding error messages to a dictionary given a key

Why do remote US companies require working in the US?

Is wanting to ask what to write an indication that you need to change your story?



How to prove a sum formula about powers of 2 and 3?



The Next CEO of Stack OverflowProve: $sum_{x=0}^{n} (-1)^x {n choose x} = 0$Prove a sum formula by inductionAlternating sum of product of Fibonacci numbersInduction based on sum of $kth$ powers.How can we prove the sum of squares/cubes/etc is always a polynomial of appropriate degree?Formula for sum of first $n$ odd integersProve the theorem by binding the term and splitting the sumHow to prove this summation formula?How can I write this as a double sum?How do I prove this equation by mathematical induction?












0












$begingroup$


I discovered this yesterday and I am wondering:




  1. Is this a well known formula?

  2. How do I go about proving something like this? (Beyond induction.)


$$2^{2n} = 3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}$$



Thank you. Any thoughts on this would be greatly appreciated.



P.s: For notation purposes, when $n=0$ the sum is not evaluated.






combinatorics summation induction exact-sequence







share|cite|improve this question











$endgroup$












  • $begingroup$
    Use the geometric sum formula $sum_{k=0}^{n-1} x^k = (1-x^n) / (1-x)$
    $endgroup$
    – Jair Taylor
    Mar 17 at 7:10












  • $begingroup$
    @JairTaylor: it is unclear to me, how that helps.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:12










  • $begingroup$
    Close, but it does not explain the powers of 2. And after you add in the binomial coeficients, it explains things even less.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:16


















0












$begingroup$


I discovered this yesterday and I am wondering:




  1. Is this a well known formula?

  2. How do I go about proving something like this? (Beyond induction.)


$$2^{2n} = 3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}$$



Thank you. Any thoughts on this would be greatly appreciated.



P.s: For notation purposes, when $n=0$ the sum is not evaluated.






combinatorics summation induction exact-sequence







share|cite|improve this question











$endgroup$












  • $begingroup$
    Use the geometric sum formula $sum_{k=0}^{n-1} x^k = (1-x^n) / (1-x)$
    $endgroup$
    – Jair Taylor
    Mar 17 at 7:10












  • $begingroup$
    @JairTaylor: it is unclear to me, how that helps.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:12










  • $begingroup$
    Close, but it does not explain the powers of 2. And after you add in the binomial coeficients, it explains things even less.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:16
















0












0








0





$begingroup$


I discovered this yesterday and I am wondering:




  1. Is this a well known formula?

  2. How do I go about proving something like this? (Beyond induction.)


$$2^{2n} = 3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}$$



Thank you. Any thoughts on this would be greatly appreciated.



P.s: For notation purposes, when $n=0$ the sum is not evaluated.






combinatorics summation induction exact-sequence







share|cite|improve this question











$endgroup$




I discovered this yesterday and I am wondering:




  1. Is this a well known formula?

  2. How do I go about proving something like this? (Beyond induction.)


$$2^{2n} = 3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}$$



Thank you. Any thoughts on this would be greatly appreciated.



P.s: For notation purposes, when $n=0$ the sum is not evaluated.






combinatorics summation induction exact-sequence




combinatorics summation induction exact-sequence






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Mar 17 at 7:11







ReverseFlow

















asked Mar 17 at 7:08









ReverseFlowReverseFlow

609513




609513












  • $begingroup$
    Use the geometric sum formula $sum_{k=0}^{n-1} x^k = (1-x^n) / (1-x)$
    $endgroup$
    – Jair Taylor
    Mar 17 at 7:10












  • $begingroup$
    @JairTaylor: it is unclear to me, how that helps.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:12










  • $begingroup$
    Close, but it does not explain the powers of 2. And after you add in the binomial coeficients, it explains things even less.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:16




















  • $begingroup$
    Use the geometric sum formula $sum_{k=0}^{n-1} x^k = (1-x^n) / (1-x)$
    $endgroup$
    – Jair Taylor
    Mar 17 at 7:10












  • $begingroup$
    @JairTaylor: it is unclear to me, how that helps.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:12










  • $begingroup$
    Close, but it does not explain the powers of 2. And after you add in the binomial coeficients, it explains things even less.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:16


















$begingroup$
Use the geometric sum formula $sum_{k=0}^{n-1} x^k = (1-x^n) / (1-x)$
$endgroup$
– Jair Taylor
Mar 17 at 7:10






$begingroup$
Use the geometric sum formula $sum_{k=0}^{n-1} x^k = (1-x^n) / (1-x)$
$endgroup$
– Jair Taylor
Mar 17 at 7:10














$begingroup$
@JairTaylor: it is unclear to me, how that helps.
$endgroup$
– ReverseFlow
Mar 17 at 7:12




$begingroup$
@JairTaylor: it is unclear to me, how that helps.
$endgroup$
– ReverseFlow
Mar 17 at 7:12












$begingroup$
Close, but it does not explain the powers of 2. And after you add in the binomial coeficients, it explains things even less.
$endgroup$
– ReverseFlow
Mar 17 at 7:16






$begingroup$
Close, but it does not explain the powers of 2. And after you add in the binomial coeficients, it explains things even less.
$endgroup$
– ReverseFlow
Mar 17 at 7:16












1 Answer
1






active

oldest

votes


















3












$begingroup$

It is:
$$3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}=3^n+3^{n-1}sum_{k=0}^{n-1}left(frac43right)^{k}=3^n+3^{n-1}frac{left(frac43right)^n-1}{frac43-1}=\
3^n+(4^n-3^n)=4^n=2^{2n}.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:19












  • $begingroup$
    No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
    $endgroup$
    – farruhota
    Mar 17 at 7:20












Your Answer





StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");

StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});

function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});


}
});














draft saved

draft discarded


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3151220%2fhow-to-prove-a-sum-formula-about-powers-of-2-and-3%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown

























1 Answer
1






active

oldest

votes








1 Answer
1






active

oldest

votes









active

oldest

votes






active

oldest

votes









3












$begingroup$

It is:
$$3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}=3^n+3^{n-1}sum_{k=0}^{n-1}left(frac43right)^{k}=3^n+3^{n-1}frac{left(frac43right)^n-1}{frac43-1}=\
3^n+(4^n-3^n)=4^n=2^{2n}.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:19












  • $begingroup$
    No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
    $endgroup$
    – farruhota
    Mar 17 at 7:20
















3












$begingroup$

It is:
$$3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}=3^n+3^{n-1}sum_{k=0}^{n-1}left(frac43right)^{k}=3^n+3^{n-1}frac{left(frac43right)^n-1}{frac43-1}=\
3^n+(4^n-3^n)=4^n=2^{2n}.$$






share|cite|improve this answer









$endgroup$













  • $begingroup$
    Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:19












  • $begingroup$
    No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
    $endgroup$
    – farruhota
    Mar 17 at 7:20














3












3








3





$begingroup$

It is:
$$3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}=3^n+3^{n-1}sum_{k=0}^{n-1}left(frac43right)^{k}=3^n+3^{n-1}frac{left(frac43right)^n-1}{frac43-1}=\
3^n+(4^n-3^n)=4^n=2^{2n}.$$






share|cite|improve this answer









$endgroup$



It is:
$$3^n+sum_{k=0}^{n-1}3^{n-k-1}2^{2k}=3^n+3^{n-1}sum_{k=0}^{n-1}left(frac43right)^{k}=3^n+3^{n-1}frac{left(frac43right)^n-1}{frac43-1}=\
3^n+(4^n-3^n)=4^n=2^{2n}.$$







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered Mar 17 at 7:18









farruhotafarruhota

21.7k2842




21.7k2842












  • $begingroup$
    Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:19












  • $begingroup$
    No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
    $endgroup$
    – farruhota
    Mar 17 at 7:20


















  • $begingroup$
    Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
    $endgroup$
    – ReverseFlow
    Mar 17 at 7:19












  • $begingroup$
    No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
    $endgroup$
    – farruhota
    Mar 17 at 7:20
















$begingroup$
Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
$endgroup$
– ReverseFlow
Mar 17 at 7:19






$begingroup$
Beautiful, thank you. Also, I have to wait 4 minutes to accept an answer, -_-.
$endgroup$
– ReverseFlow
Mar 17 at 7:19














$begingroup$
No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
$endgroup$
– farruhota
Mar 17 at 7:20




$begingroup$
No hurry, you should wait longer to see if others can provide more elegant solutions. Good luck.
$endgroup$
– farruhota
Mar 17 at 7:20


















draft saved

draft discarded




















































Thanks for contributing an answer to Mathematics Stack Exchange!


  • Please be sure to answer the question. Provide details and share your research!

But avoid …



  • Asking for help, clarification, or responding to other answers.

  • Making statements based on opinion; back them up with references or personal experience.


Use MathJax to format equations. MathJax reference.


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




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3151220%2fhow-to-prove-a-sum-formula-about-powers-of-2-and-3%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown





















































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown

































Required, but never shown














Required, but never shown












Required, but never shown







Required, but never shown







-

Popular posts from this blog

Nidaros erkebispedøme

Image
For det lutherske bispedømet, sjå Nidaros bispedøme. Nidaros erkebispedøme var eit katolsk erkebispedøme med sete i Nidaros. Det vart oppretta av kong Olav Kyrre mellom 1070 og 1080 som bispedøme. Bispedømet vart opphøgd til erkebispedøme i 1153. Erkebispedømet vart innstifta av pave Anastasius IV i 1154, og var i funksjon fram til reformasjonen i 1537. Erkebiskopen sitt hovudsete var Erkebispegarden, rett ved domkyrkja Nidarosdomen. Frå 1152 til 1154 var Nicholas Breakspear i Skandinavia som paveleg utsending, og organiserte opprettinga av erkesetet. Innhaldsliste 1 Utbreiing og jurisdiksjon 2 Overhyrdar 2.1 Lydbiskopar 3 Biskopar 3.1 Biskopar 3.2 Erkebiskopar Utbreiing og jurisdiksjon | Kart over den nordiske kyrkjeprovinsen (1153-1387) Erkebispedømet omfatta i tillegg til Noreg øyane i vest i Noregsveldet. Nidaros var eit metropolitanerkebispedøme med fylgjande underliggjande bispedøme suffraganar: Bjørgvin bispedøme Oslo bispedøme Hamar bispedøme Stavanger bispedøme Kirkjub
Read more

Birsay

Image
Tettstaden Birsay på Orknøyane. Ruinane av jarleplasset om lag midt i biletet. (Fotografi frå 2012) Utsikt mot Birsay og Brough of Brisay (fotografi frå 2009) Birsay er ein tettstad i eit sokn (parish) med same namn, nordvest på øya Mainland [Note 1] på Orknøyane, om lag 31 km frå Kirkwall. Staden er omtala fleire stader i Orknøyingasoga [1] , der han er kalla «Byrgesherad» (norrønt: Birgisherað ). Namnet skriv seg truleg frå elvenamnet Birgisá [2] . Innhaldsliste 1 Den gamle hovudstaden 2 Bispesetet 3 Jarlepalasset 4 Kjelder og utfyllande opplysningar 4.1 Notar 4.2 Fotnotar 4.3 Bakgrunnsstoff Den gamle hovudstaden | Hovudartikkelen for dette emnet er Torfinn Sigurdsson . Soknet er i dag i hovudsak ei jordbrukskommune men er mest kjend for dei mange kulturminna i området. Tettstaden Birsay blir gjerne omtala som «den gamle hovudstaden på Orknøyane» [3] , og ein viser då gjerne til Orknøyingasoga (kap.31) som fortel at orknøyjarlen Torfinn Sigurdsson den mektige «hadde jamleg
Read more

Where did Arya get these scars? Unicorn Meta Zoo #1: Why another podcast? Announcing the arrival of Valued Associate #679: Cesar Manara Favourite questions and answers from the 1st quarter of 2019Why did Arya refuse to end it?Has the pronunciation of Arya Stark's name changed?Has Arya forgiven people?Why did Arya Stark lose her vision?Why can Arya still use the faces?Has the Narrow Sea become narrower?Does Arya Stark know how to make poisons outside of the House of Black and White?Why did Nymeria leave Arya?Why did Arya not kill the Lannister soldiers she encountered in the Riverlands?What is the current canonical age of Sansa, Bran and Arya Stark?

Image
How to translate "red flag" into Spanish? France's Public Holidays' Puzzle How did Elite on the NES work? Where can I find how to tex symbols for different fonts? In search of the origins of term censor, I hit a dead end stuck with the greek term, to censor, λογοκρίνω How to check if string is entirely made of same substring? What's called a person who works as someone who puts products on shelves in stores? What helicopter has the most rotor blades? Did war bonds have better investment alternatives during WWII? Page Layouts : 1 column , 2 columns-left , 2 columns-right , 3 column Marquee sign letters Why did Israel vote against lifting the American embargo on Cuba? What *exactly* is electrical current, voltage, and resistance? Mechanism of the formation of peracetic acid Does Prince Arnaud cause someone holding the Princess to lose? Bright yellow or light yellow? What is /etc/mtab in Linux? What is the ongoing value of the Kanban
Read more