Question

P= (1+1/1x3)x(1+1/2x4)x(1+1/3x5)x......x(1+1/49x51)+2/51 ta được phân số tối giản .........

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Answer 1

\(\left(1+\dfrac{1}{1.3}\right)\left(1+\dfrac{1}{2.4}\right)\left(1+\dfrac{1}{3.5}\right)...\left(1+\dfrac{1}{49.51}\right)\)+\(\dfrac{2}{51}\)

=\(\dfrac{4}{1.3}.\dfrac{9}{2.4}.\dfrac{16}{3.5}.....\dfrac{2500}{49.51}\)+\(\dfrac{2}{51}\)

=\(\dfrac{2^2}{1.3}.\dfrac{3^2}{2.4}.\dfrac{4^2}{3.5}.....\dfrac{50^2}{49.51}\)+\(\dfrac{2}{51}\)

=\(\dfrac{\left(2.3.4.....50\right)\left(2.3.4.....50\right)}{\left(1.2.3.....49\right)\left(3.4.....51\right)}\)+\(\dfrac{2}{51}\)

=\(\dfrac{\left(2.3.4.....49\right).50.2.\left(3.4.5.....50\right)}{1.\left(2.3.4.....49\right)\left(3.4.5.....50\right).51}\)+\(\dfrac{2}{51}\)

=\(\dfrac{50.2}{1.51}\)+\(\dfrac{2}{51}\)=\(\dfrac{100}{51}\)+\(\dfrac{2}{51}\)=\(\dfrac{102}{51}\)=2