Godfavour Godfavour
  • 08-11-2017
  • Mathematics
contestada

show that sum k=1/2n(n+1) from k=1 to n

Respuesta :

LammettHash
LammettHash LammettHash
  • 08-11-2017
Suppose the value of the sum is [tex]S[/tex]:

[tex]S=\displaystyle\sum_{k=1}^nk[/tex]

So

[tex]S=1+2+3+\cdots+(n-2)+(n-1)+n[/tex]

but also

[tex]S=n+(n-1)+(n-2)+\cdots+3+2+1[/tex]

That is,

[tex]S=\displaystyle\sum_{k=1}^n(n-k+1)[/tex]

Adding these together, we have

[tex]2S=\displaystyle\sum_{k=1}^n(k+n-k+1)=\sum_{k=1}^n(n+1)=(n+1)\sum_{k=1}^n1=n(n+1)[/tex]
[tex]\implies S=\dfrac{n(n+1)}2[/tex]
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