What is the formula for calculating the sum of ceiling values for two integers?

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Discussion Overview

The discussion revolves around finding a formula for calculating the sum of ceiling values for two integers, specifically focusing on the expression ceiling[(n+m)/2] + ceiling[(n-m+1)/2]. The scope includes mathematical reasoning and case analysis based on the parity of the integers involved.

Discussion Character

  • Mathematical reasoning, Technical explanation, Debate/contested

Main Points Raised

  • One participant proposes the formula ceiling[(n+m)/2] + ceiling[(n-m+1)/2] for integers 'n' and 'm'.
  • Another participant suggests considering two cases based on the parity of 'n' and 'm': both even or both odd, and one odd and one even.
  • A further contribution emphasizes splitting the analysis into cases to explicitly find the ceilings for each scenario.
  • One participant requests more detailed guidance on how to analyze the cases of even and odd parities separately.
  • Another participant discusses how to express even and odd integers and suggests substituting these expressions into the cases to evaluate the sum.
  • It is noted that the sum or difference of two even numbers is even, the sum or difference of two odd numbers is even, and the sum or difference of one odd and one even number is even, which may impact the evaluation of the formula.
  • One participant highlights that in any case, one of the terms (n+m)/2 or (n-m+1)/2 will be an integer while the other will be a half-integer.

Areas of Agreement / Disagreement

Participants appear to agree on the need to analyze the parity of the integers involved, but there is no consensus on the final evaluation or simplification of the formula, as multiple approaches are being discussed.

Contextual Notes

The discussion does not resolve the mathematical steps required to evaluate the formula fully, and the implications of parity on the ceiling function remain open for further exploration.

rayukpant
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For integers 'n' and 'm', find the value of ceiling[(n+m)/2] + ceiling[(n-m+1)/2].
 
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Consider the two cases separately, of
1. Either both even or both odd and
2. One odd the other even.
 
Just split it into cases.

Either
(1) m and n have the same parity (either both odd or both even); or
(2) m and n have different parities (one odd and the other even).

In both cases, you can explicitly find the ceilings in question.
 
can you please tell me more in detail as to how to consider those two separately with even and odd parities
 
How can you express an even integer? Odd integer? Once you have expressions for each, go through each case and plug them in. After you do this, you can evaluate explicitly what the answer is.
 
The sum of difference of two even numbers is even, the sum or difference of two odd numbers is even. The sum or difference of one odd and one even number is even. Fit those into your formula. The crucial point is that in any case one of (n+m)/2 and (n- m+ 1)/2 is an integer and the other is a half integer.
 

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