Question

Va rog am nevoie de ea urgent !
1 a)Fie numerele rationale pozitiv
a=1 supra 3+ 3 supra 5 + 5 supra 7+.....+2015 supra 2017
b= 2 supra 3 + 2 supra 5 +2 supra 7 +.......+2 supra 2017.
Calculati a+b

Answer 1

$a = \sum\limits_{k=1}^{1008} \dfrac{2k-1}{2k+1} \\ \\ b =\sum\limits_{k=1}^{1008} \dfrac{2}{2k+1} \\ \\ a + b = \sum\limits_{k=1}^{1008} \dfrac{2k-1}{2k+1} + \sum\limits_{k=1}^{1008} \dfrac{2}{2k+1} = \\=\sum\limits_{k=1}^{1008} \Big(\dfrac{2k-1}{2k+1} + \dfrac{2}{2k+1}\Big) = \\ = \sum\limits_{k=1}^{1008} \dfrac{(2k-1) +2}{2k+1} = \\ = \sum\limits_{k=1}^{1008} \dfrac{2k+1}{2k+1} = \\ = \sum\limits_{k=1}^{1008} 1 = \\ \\ = 1008$