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How can we prove the following inequality?

$$2^{135}+3^{133}<4^{108}$$

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    $\begingroup$ Why don't you just use a calculator and compare the results? $\endgroup$ Aug 16, 2013 at 16:27
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    $\begingroup$ can he calculate $2^{135}$? $\endgroup$ Aug 16, 2013 at 16:32
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    $\begingroup$ @Vijay: If the question was " Prove the inequality $2^4+3^4<4^4$ ", would you still say that using a calculator is an invalid response? I don't think I would. $\endgroup$ Aug 16, 2013 at 16:39
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    $\begingroup$ by prove, you mean prove by analytic methods. using a calculator is known as 'verification'-checking your hypothesis $\endgroup$
    – dajoker
    Aug 16, 2013 at 16:41
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    $\begingroup$ @VijayRaghavan: I disagree. Proving is verifying correctness. If explicit calculation can confirm this, then this a perfectly valid proof. On the other hand, this is most likely not what the OP actually wants. $\endgroup$
    – Tomas
    Aug 16, 2013 at 18:40

11 Answers 11

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First,

$$2^{135}=2^7\cdot2^{128}=2^7\cdot(2^8)^{16}<(2^8)^{17}=256^{17}\;.$$

Next,

$$3^{133}=3^3\cdot3^{130}=3^3\cdot(3^5)^{26}=3^3(243)^{26}<3^3(256)^{26}=27\cdot256^{26}\;.$$

Thus,

$$2^{135}+3^{133}<256^{17}+27\cdot256^{26}=(1+27\cdot256^9)256^{17}<256^{10}\cdot256^{17}=256^{27}=4^{108}\;.$$

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  • $\begingroup$ Aha! Very nice. $\endgroup$ Aug 16, 2013 at 16:43
  • $\begingroup$ Excellent. Way to go!!! $\endgroup$
    – dajoker
    Aug 16, 2013 at 16:44
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    $\begingroup$ Minor simplification: $27\cdot 256^{26}<2^5\cdot2^{8\cdot 26}=2^{213}$, thus $2^{133}+3^{133}<2\cdot 3^{133}<2\cdot 2^{213}=2^{214}=4^{107}$. This is quite close to the "true" exponent $\approx 105.4$. $\endgroup$ Aug 16, 2013 at 17:06
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While there are several more clever answers up there, I couldn't resist posting this answer.

2^135 =                          43556142965880123323311949751266331066368
3^133 =   2865014852390475710679572105323242035759805416923029389510561523
4^108 = 105312291668557186697918027683670432318895095400549111254310977536

So even by eye, you can confirm that $2^{135}+3^{133}<4^{108}$.

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    $\begingroup$ Since the inequality is so far, and you can do it "by eye", perhaps we could consider using $\log_{10}$ along with estimates for the $\log_{10} 2, \log_{10} 3$ (and then $\log_{10} 4$ follows). $\endgroup$ Aug 16, 2013 at 19:23
  • $\begingroup$ @GeorgeV.Williams Yeah, sure. $\endgroup$
    – Ali
    Aug 16, 2013 at 19:54
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    $\begingroup$ Hm, one possible concern with eyeballing such long strings is that, without counting, are we sure that the string with less horizontal length has the least number of digits? Depending on the formatting, some characters may take up less space than others. It appears things are pretty equal here though. $\endgroup$ Aug 16, 2013 at 19:59
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    $\begingroup$ @Jonas: how's that? Ali, I hope you don't mind the edit... $\endgroup$
    – Mike F
    Aug 17, 2013 at 0:19
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    $\begingroup$ @JonasMeyer This would be an issue in written form or with usual font but Ali made sure to use a monospace one, which has the same horizontal and vertical space allocation for any kind of symbol. $\endgroup$ Aug 17, 2013 at 8:27
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Using $3^5 < 2^8$ we have $3^{130}<2^{208}$.

$$2^{135}+3^{133}< 2^{135}+3^{3}2^{208}<2^{208}+3^{3}2^{208}=(1+27)4^{104}<4^4 \cdot 4^{104} $$

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Note that $2^6<3^4$, so $$2^{135}=2^{129}2^6<2^{129}3^4<3^{129}3^4=3^{133}.$$ Therefore $$2^{135}+3^{133}<3^{133}+3^{133}=2\cdot 3^{133}<3\cdot 3^{133}=3^{134}.$$

Next note that $3^5<2^8$, so $$3^{134}<2^{134\cdot (8/5)}<2^{215}<2^{216}=4^{108}.$$

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Hint: log$_2(3)\approx 1.5850$… Consider writing $3^{133}$ as a power of $2$ and factoring.

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  • $\begingroup$ It is enough to use $\log_2(3)<8/5$, i.e., $3^5<2^8$. $\endgroup$ Aug 16, 2013 at 16:58
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Note that $4=2^2$ and $\frac {3^4}{4^4} \lt 3^{-1}$ - also that $108=4\times 27$

On dividing through by $4^{108}$ and working with the expression we find $$\frac {2^{135}}{4^{108}}+\frac {3^{133}}{4^{108}}=2^{-81}+3^{25}\left( \frac{3^4}{4^4}\right)^{27}\lt 2^{-81} + 3^{-2}\lt \frac 12+\frac 13\lt 1$$

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Another one that uses the $$ 3^5 < 2^8 = 4^4 $$

and the easy to prove: $$2^{135} + 3^{133} < 3^{135}$$

Then:

$$2^{135} + 3^{133} < 3^{135} = (3^5)^{27} < (4^4)^{27} = 4^{108} $$

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$$3^{135} = 3^{5 \cdot 27} < 4^{4 \cdot 27} = 4^{108} \implies {2^{135}+3^{133}} < 3^{134} < {4^{108}}\;.$$

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Take logarithms, and don't worry about even large differences:

$$ \begin{eqnarray} 135\log 2&\approx &93.57\\ 133\log 3&\approx &146.12\\ 135\log 2&<&133\log 3\\ 108\log 4&\approx &149.72\\ \log[2^{135} + 3^{133}]&< &\log (2 \cdot 3^{133})\\ \log (2 \cdot 3^{133}) &\approx &146.81 \end{eqnarray} $$

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i would like different method,first let us do following things

$2^{135}=2^{27}*2^{108}$

and $ 3^{133}=3^{25}*3^{108}$

so we have

$2^{27}*2^{108}+3^{25}*3^{108}<4^{108}$

can't proceed from this?

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    $\begingroup$ It is unclear to me what you would suggest doing with this reformulation. $\endgroup$ Aug 16, 2013 at 16:56
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    $\begingroup$ dato, I do not understand what your comment means. If "way" means a way to solve the problem, I don't know what way you are suggesting. There is a reformulation of the inequality, but I don't understand how it is being suggested that this reformulation could be used to prove that it is true. $\endgroup$ Aug 16, 2013 at 17:05
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    $\begingroup$ That is a different inequality, which in the case of $n=108$, is much weaker. It is unclear to me how it would show a solution to this problem. I have no doubt that one could start some solution with this observation, but it does not appear to be indicated how in this answer. $\endgroup$ Aug 16, 2013 at 17:08
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    $\begingroup$ ok if it is bad,you can downvote or i can delete it,no problem $\endgroup$ Aug 16, 2013 at 17:11
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    $\begingroup$ @JonasMeyer The solution I posted deals with the $3$ part of the solution in a manner compatible with this, and the $2$ part is easily adaptable. $\endgroup$ Aug 16, 2013 at 19:05
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$9=3^2>2^3=8 \implies 3^{90}>2^{135}$

$\begin{align}256=4^4>3^5=243 &\implies \\ 4^{108} &>3^{135} \\ &>3^{90}+3^{133}\\ &>2^{135}+3^{133}\end{align}$

Clearly we are a long way from having a close inequality there. The same approach would justify $2^{201}+3^{134}<4^{108}$

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