Rút gọn phân số sau : $\frac{1^3+2^3+3^3+……+100^3}{1.5+2.8+3.11+…..+100.302}$ 23/10/2021 Bởi Isabelle Rút gọn phân số sau : $\frac{1^3+2^3+3^3+……+100^3}{1.5+2.8+3.11+…..+100.302}$
Bài LàmTS= 1^3+2^3+3^3+..+100^3 = (1 + 2 + 3 + 4 + … + 100)^2 = [100. (100 + 1)]^2 = (100. 101)^2 = 101. 101. 100. 100.MS= 1.5+2.8+3.11+…+100.302. => 1.(3.1 + 2) + 2.(3.2 + 2) + 3.(3.3 + 2) + 4.(3.4 + 2) + … + 100.(3.100 + 2).=> 3.(1^2 + 2^2 + 3^2 + 4^2 + … + 100^2) + 2.(1 + 2 + 3 + 4 + 100).=> 3. [1.(2-1) + 2.(3 – 1) + 3.(4 – 1) + 4.(5 – 1) + … + 100.(101 – 1)] + 2.100.(100 + 1): 2.=> 3.[(1.2 + 2.3 + 3.4 + 4.5 + … + 100.101) – (1 + 2 + 3 + 4 + … + 100)] + 100.101.=> 3.[(1.2.3 + 2.3.3 + 3.4.3 + 4.5.3 + … + 100.101.3) – 100.(100 + 1) : 2] + 100.101.=> 3.[(1.2.(3 – 0) + 2.3(4-1) + 3.4.(5 – 2) + …. + 100. 101. (102 – 99) – 50.101] + 100.101.=> 3.[(1.2.3 – 0.1.2 + 2.3.4 – 1.2.3 + 3.4.5 – 2.3.4 + …. + 100.101.102 – 99.100.101 – 50.101)] + 100.101.=> 3.[100.101.102 – 50.101] + 100.101.=> 101. 30550.Vậy 100^2.101/30550. #Chúcbạnhọctốt! #Tử Thiên. Bình luận
Bài Làm
TS= 1^3+2^3+3^3+..+100^3 = (1 + 2 + 3 + 4 + … + 100)^2 = [100. (100 + 1)]^2 = (100. 101)^2 = 101. 101. 100. 100.
MS= 1.5+2.8+3.11+…+100.302.
=> 1.(3.1 + 2) + 2.(3.2 + 2) + 3.(3.3 + 2) + 4.(3.4 + 2) + … + 100.(3.100 + 2).
=> 3.(1^2 + 2^2 + 3^2 + 4^2 + … + 100^2) + 2.(1 + 2 + 3 + 4 + 100).
=> 3. [1.(2-1) + 2.(3 – 1) + 3.(4 – 1) + 4.(5 – 1) + … + 100.(101 – 1)] + 2.100.(100 + 1): 2.
=> 3.[(1.2 + 2.3 + 3.4 + 4.5 + … + 100.101) – (1 + 2 + 3 + 4 + … + 100)] + 100.101.
=> 3.[(1.2.3 + 2.3.3 + 3.4.3 + 4.5.3 + … + 100.101.3) – 100.(100 + 1) : 2] + 100.101.
=> 3.[(1.2.(3 – 0) + 2.3(4-1) + 3.4.(5 – 2) + …. + 100. 101. (102 – 99) – 50.101] + 100.101.
=> 3.[(1.2.3 – 0.1.2 + 2.3.4 – 1.2.3 + 3.4.5 – 2.3.4 + …. + 100.101.102 – 99.100.101 – 50.101)] + 100.101.
=> 3.[100.101.102 – 50.101] + 100.101.
=> 101. 30550.
Vậy 100^2.101/30550.
#Chúcbạnhọctốt!
#Tử Thiên.