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Let's have the following fractions.

$ \frac{752}{375} + \frac{754}{376}+ \frac{756}{377} + \frac{758}{378}+ \frac{760}{379} \approx 10\times(\frac{5}{375}+1)^{1/5}$

$\frac{752}{375}+ \frac{754}{376}+ \frac{756}{377}+ \frac{758}{378}+ \frac{760}{379}+ \frac{762}{380} \approx 12\times(\frac{6}{375}+1)^{1/6}$

$\frac{752}{375}+ \frac{754}{376}+\frac{756}{377}+ \frac{758}{378}+ \frac{760}{379}+ \frac{762}{380}+ \frac{764}{381} \approx 14\times(\frac{7}{375}+1)^{1/7}$

What is the explanation for these almost equalities?

Transcription of math: 752/375 + 754/376 + 756/377 + 758/378 + 760/379 ≈ 10 * (5/375 + 1)^(1/5), then 752/375 + 754/376 + 756/377 + 758/378 + 760/379 + 762/380 ≈ 12 * (6/375 + 1)^(1/6), then 752/375 + 754/376 + 756/377 + 758/378 + 760/379 + 762/380 + 764/381 ≈ 14 * (7/375 + 1)^(1/7)

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  • $\begingroup$ Please, when you ask [mathematics] questions, use the broad tag as well as any specific sub-field tag. Also, what is the question here exactly? Do you want a mathematical justification for the "almost equalities"? In the form of an equation/proof? $\endgroup$ – bobble Nov 25 '20 at 22:53
  • $\begingroup$ Any justification will do. $\endgroup$ – Vassilis Parassidis Nov 25 '20 at 22:59
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    $\begingroup$ Please don't use * in MathJax to represent multiplication. Use \times instead. $\endgroup$ – Bubbler Nov 25 '20 at 23:19
  • $\begingroup$ Should solvers be concerned with the way you don't define 'almost'? $\endgroup$ – Cotton Headed Ninnymuggins Nov 25 '20 at 23:28
  • $\begingroup$ @ cotton, you should not be concerned. $\endgroup$ – Vassilis Parassidis Nov 26 '20 at 0:33
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They can be justified by viewing the left and right sides of each (approximate) equality as

the arithmetic mean and the geometric mean of some very close numbers, respectively.

To see this, let's focus on the first line and transform it a bit.

Divide both sides by 2: $$ \frac{376}{375} + \frac{377}{376} + \frac{378}{377} + \frac{379}{378} + \frac{380}{379} \approx 5 \times \left(\frac{380}{375}\right)^{1/5} $$

and then by the number of terms on the left side: $$ \frac{\frac{376}{375} + \frac{377}{376} + \frac{378}{377} + \frac{379}{378} + \frac{380}{379}}{5} \approx \left(\frac{380}{375}\right)^{1/5} $$

Since $$ \frac{376}{375} \times \frac{377}{376} \times \frac{378}{377} \times \frac{379}{378} \times \frac{380}{379} = \frac{380}{375} $$ it follows that the left side is the arithmetic mean of the five fractions, and the right side is their geometric mean. The other two lines can be justified in the similar fashion.

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