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Proximo Ultimo Sum (Posted on 2011-11-16) Difficulty: 3 of 5
Let us consider a sequence whose first term is 1.2 - and, the numbers appearing respectively to the right and the left of the decimal point of a given term are swapped and this number is added to a given term to obtain the next term.

For example, if a term is 46.78 then the next term will be 46.78+ 78.46 = 125.24.

For some of the terms in the sequence, the numbers appearing before the decimal point and the numbers appearing after the decimal point are congruent. Considering the first 30 terms, how many terms do have this property?

Note: Terms like 792.792 are deemed to have the desired property. However terms like 3870.387 do not possess this property as 3870 is not equal to 387.

See The Solution Submitted by K Sengupta    
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The easy way is turning out to be hard | Comment 1 of 5
The sequence is easy to generate with a calculator, but it is a bit tedious.

I'm trying instead to generate the sequence using excel.  It is hard to work around its rounding issues.   

When you subtract the integer part of a number from the number to isolate the decimal part you get a rounding error around the tenth decimal point.  This is remedied by rounding the decimal after converting it to a whole number.

The bigger problem is on the 20th term:  132385.3238
Excel is splitting this into 132385 and .32384
I cannot figure out where this error is coming from or how to fix it.

More interesting than the property the problem is asking about is how the terms grow.   It appears to be roughly exponential.

  Posted by Jer on 2011-11-16 11:22:03
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