Question

rog sediozitate va roggg ex 4​

Answer 1

folosim formula sumei Gauss:

$\boxed {1 + 2 + 3 + ... + n = \dfrac{n \cdot (n + 1)}{2}}$

pentru:

$2+4+6+...+2004 = 2 \cdot (1+2+3+...+1002) =$

$= 2 \cdot \dfrac{1002 \cdot 1003}{2} = 1002 \cdot 1003$

și formula:

$\boxed{\dfrac{1}{n \cdot (n + 1)} = \dfrac{1}{n} - \dfrac{1}{n + 1}}$

pentru:

$\dfrac{1}{1 \cdot 2} + \dfrac{1}{2 \cdot 3} + \dfrac{1}{3 \cdot 4} + \dfrac{1}{4 \cdot 5} + ... + \dfrac{1}{1002 \cdot 1003} =$

$= \dfrac{1}{1} - \dfrac{1}{2} + \dfrac{1}{2} - \dfrac{1}{3} + \dfrac{1}{3} - \dfrac{1}{4} + \dfrac{1}{4} - \dfrac{1}{5} + ... + \dfrac{1}{1002} - \dfrac{1}{1003}$

reducem termenii asemenea:

$= \dfrac{1}{1} - \dfrac{1}{1003} = \dfrac{1003 - 1}{1003} = \dfrac{1002}{1003}$

atunci putem scrie:

$n = 1002 \cdot 1003 \cdot \dfrac{1002}{1003} = \bf {1002}^{2}$

=> n este pătrat perfect

q.e.d.