Thursday, January 31, 2019

discrete mathematics - Sum of Arithmetic progression problem


I have following progression:



$2n+(2n-1)+\cdots+n$



which is equivalent to:



$n+(n+1)+\cdots+2n$



The answer says:




You match $2n-i$ with $n+i$ for all $i=0..n$ Therefore you have $(n+1)$ terms of $3n$, so the sum of the $2$ exactly same sequence is $3n(n+1)$ and therefore the sum of 1 sequence is $1.5n(n+1)$


Or you can just apply the formula for the sum of Arithmetic progression, please refer to the wiki page https://en.wikipedia.org/wiki/Arithmetic_progression



Thus, I tried applying sum of arithmetic progression since I need to make it "$1.5n(n+1)$". but when I apply sum of arithmetic progression, it gives me different result like "$\frac n2(2n+(n-1)*1) = 1.5n^2-0.5n$"


How can I get "$1.5n(n+1)$"?


Answer



The sum of an artihmetic progression is (first term + last term)*(number of terms)/2.


Here :


  • first term=n,


  • last term=2n,

  • number of terms=n+1

Applying the formula, $Sum=\frac{(n+2n)*(n+1)}{2}$ reaches your desired result...


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