\(1.3.5....99=\dfrac{51}{2}.\dfrac{52}{2}....\dfrac{100}{2}\)
Nhân cả hai vế với \(1.2...50.2^{50},\) ta được:
Vế 1:
\(1.3.5....99.1.2...50.2^{50}=1.3.5...99.2.2.2.2.1.2...50\)
\(=1.3.5...99.1.2.2.2.2.3.2.4...2.50\)
\(=1.3.5...99.2.4...10=1.2.3.4.5...100\) ( 1 )
Vế 2:
\(\dfrac{51}{2}.\dfrac{52}{2}....\dfrac{100}{2^{50}}.1.2.3....50=\dfrac{51}{2}.\dfrac{52}{2}....\dfrac{100}{2}.2.2...1.2.3...50\)
\(=\left(\dfrac{51}{2}\right).2.\left(\dfrac{52}{2}\right).2...\left(\dfrac{100}{2}\right).2...1.2.3...50\)
Rút gọn ta sẽ được:
\(51.52.52....100.1.2.3...50\) ( 2 )
Từ ( 1 ) và ( 2 ) \(\Rightarrow1.3.5...99=\dfrac{51}{2}.\dfrac{52}{2}....\dfrac{100}{2}\)
