Question

Se considera numerele m, n aparține N*. Sa se arate ca:

$(m + 1)( \frac{m}{2} + 1) \times ... \times ( \frac{m}{n} + 1) = (n + 1)( \frac{n}{2} + 1) \times ... \times ( \frac{n}{m} + 1).$

Answer 1

Aducand la acelasi numitor in fiecare paranteza si inmultind, egalitatea devine:
$\it \frac{(m+1)(m+2)(m+3) \cdot ... \cdot (m+n)} {2 \cdot 3 \cdot ... \cdot n} = \frac{(n+1)(n+2)(n+3) \cdot ... \cdot (n+m)}{2~. ~ 3 \cdot ... \cdot m}$
de unde obtinem
$\it 2 \cdot 3 \cdot... ~\cdot m(m+1)~(m+2) \cdot ... ~ \cdot (m+n) =2 \cdot 3 \cdot... ~\cdot n(n+1)~(n+2) \cdot.. ~\cdot (n+m),$
relatie adevarata, deoarece in ambii membri avem produsul tuturor numerelor de la 2 pana la (m + n).

Answer 2

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