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Is it possible to write a mathematical formula for the following sequence: $4.2, 6.4, 8.6, 10.8, 12.10, 14.12, 16.14$...etc

looking at the components of the numbers separately, it is easily noticed that the integer part of the number increases by $2$ and that the decimal part of the number increases by $2$ as well.

When trying to find a pattern looking at the rational numbers as a whole, not at its individual components, there is no clear pattern that would help create a universal formula for this numbers. Is there any mathematical reason explaining why this sequence es possible or not possible ?

MM8
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user27407
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  • It starts like +2.2 then it goes +2.02,it probably goes +2.002 after. Though you can't tell much from just a few numbers. – kingW3 Feb 02 '17 at 22:12
  • Writing a term by term sequence is not straightforward, since the rule changes as you go along. However, if you write every 5th term, there's a clear rule. 4.2 → 14.12 → 24.112... – Kaynex Feb 02 '17 at 22:16
  • You mean arithmophiles, I suppose? – Bernard Feb 02 '17 at 22:34
  • What is keeping you from clicking the check mark to Accept one of these fine answers? – BruceET Sep 26 '17 at 23:12

4 Answers4

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Hint: the decimal representation of a positive integer $n$ has $\lfloor\log_{10}n\rfloor+1$ digits (see e.g. here).

Then $\cfrac{2\,n}{10^{\lfloor\log_{10}(2n)\rfloor+1}}$ generates the sequence $0.2, 0.4,0.6,0.8,0.10,0.12,0.14\cdots$

Community
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dxiv
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Sure, it is possible to write a formula, but it's ugly:

$$ a_n = 2 + \left(2 + 2\times10^{-1-\lfloor \log_{10}(2n) \rfloor} \right)n. $$

The "ugliness" comes because of the way you "look at each of the components of the number separately," in particular the way you manipulate the fractional component in order to treat it as an integer. That step of the "find the next term" process is defined in terms of a lexical operation on the base-ten representation of the number rather than a numeric operation on the number itself, so in order to turn it into a numerical operation we have to do some calculations to figure out how many powers of ten we would need to multiply by in order to turn the fractional part into an integer. Hence we have a logarithm and also rounding via the "floor" function, $\lfloor x \rfloor$.

As pointed out in a comment, what is "ugly" is a matter of opinion.

David K
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The sequence is quite "possible" as generating the integer and fractional parts is immediate, and you just need to "glue" them. This requires to know the number of significant digits in the fractional part, which you can then shift by dividing by a power of $10$.

The number of significant digits of a number is simply given by its base-$10$ logarithm rounded to the next integer.

Example:

$18.16$ is obtained as $18+16\cdot10^{-2}$ where $2=\lfloor\log_{10}16\rfloor+1=\lfloor1.2041\cdots\rfloor+1$.

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Notice that the number of digits in a number is $\lfloor \log_{10} x\rfloor + 1$, so therefore the formula to put a decimal before any number is $$\frac{ x}{10^{\lfloor \log_{10} x\rfloor + 1}}$$ Now, in your case, we are putting a decimal before all the even integers, so we change $x$ to $2x$ $$\frac{ 2x}{10^{\lfloor \log_{10} 2x\rfloor + 1}}$$ And finally, we add the integer part, $2x$, and then the sequence "offset" of $2$, and factor out a $2$.

$$a_n=2\big( \frac{ x}{10^{\lfloor \log_{10}2 x\rfloor + 1}}+x+1\big)$$

Nico A
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