Question

[Đặng Khôi Nguyên]

Tính nhanh1.5+2.6+3.7+...+34.38

Answer 1

[Trương Nhật Minh]

\(S=1.5+2.6+3.7+...+34.38\\ =1\left(2+3\right)+2\left(3+3\right)+3\left(4+3\right)+...+34\left(35+3\right)\\ =1.2+2.3+3.4+...+34.35+\left(3+6+9+...+102\right)\\ \)

\(S_1=1.2+2.3+3.4+...+34.35\\ =\dfrac{1}{3}\left(1.2.3+2.3.3+3.4.3+...+34.35.3\right)\\ =\dfrac{1}{3}\left(1.2.3+2.3\left(4-1\right)+3.4.\left(5-2\right)+...+34.35.\left(36-33\right)\right)\\ =\dfrac{1}{3}\left(1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+34.35.36-33.34.35\right)\\ =\dfrac{1}{3}\left(34.35.36\right)=12.34.35\)

\(S_2=3+6+9+...+102\\ =\dfrac{3+102}{2}\left(\dfrac{102-3}{3}+1\right)=\dfrac{105}{2}.34=105.17\)

\(S=S_1+S_2=12.34.35+105.17=16065\)

Answer 2

[Lương Thị Vân Anh]

1 . 5 + 2 . 6 + 3 . 7 + ... + 34 . 38

= 1( 2 + 3 ) + 2( 3 + 3 ) + 3( 3 + 4 ) + ... + 34( 35 + 3 )

= 1 . 2 + 1 .3 + 2 . 3 + 2 .3 + 3 . 3 + 3 . 4 + ... + 34 . 35 + 34 . 3

= ( 1 . 2 + 2 . 3 + 3 . 4 + ... + 34 . 35 ) + ( 1 . 3 + 2 . 3 + 3 . 3 + ... + 34 . 3 )

Đặt A = 1 . 2 + 2 . 3 + 3 . 4 + ... + 34 . 35 

3A = 1 . 2 . 3 + 2 . 3 . 3 + 3 . 4 . 3 + ... + 34 . 35 . 3

3A = 1 . 2 . 3 + 2 . 3 ( 4 -1 ) + 3 . 4 ( 5 - 2 ) + ... + 34 . 35 ( 36 - 33 )

3A  = 1 . 2 . 3 + 2 . 3 . 4 - 1 . 2 . 3 + 3 . 4 . 5 - 2 . 3 . 4 + ... + 34 . 35 . 36 - 33 . 34 . 35

3A = 34 . 35 . 36

A = 14280

B = 1 . 3 + 2 . 3 + 3 . 3 + ... + 34 . 3 

B = 3 ( 1 + 2 + 3 + ... + 34 )

B = 3 [ ( 34 + 1 ) . 34 + 1 ]

B = 3 ( 35 . 34 + 1 )

B = 3 . 1191

B = 3573

Vậy 1 . 5 + 2 . 6 + 3 . 7 + ... + 34 . 38 = 14280 + 3573 = 17853