Question

Cho A= 1.3.5.7....99

B=(51/2).(52/2).....(100/2)

Chứng tỏ A=B

Dấu gạch ở trên là phần

Answer 1

Ta có:

\(B=\dfrac{51}{2}.\dfrac{52}{2}.\dfrac{53}{2}...\dfrac{100}{2}=\dfrac{51.52.53...100}{2^{50}}\)

\(=\dfrac{\left(51.52.53..100\right)\left(1.2.3.4...50\right)}{2^{50}\left(1.2.3.4...50\right)}\)

\(=\dfrac{1.2.3.4.5.6...100}{\left(2.1\right)\left(2.2\right)\left(2.3\right)...\left(2.50\right)}\)

\(=\dfrac{1.2.3.4.5.6...100}{2.4.6.8.10...100}=\dfrac{\left(1.3...99\right)\left(2.4...100\right)}{2.4.6...100}\)

\(=1.3.5.7.99=A\)

Vậy \(A=B\) (Đpcm)