Question

计算：(1-

3

2×4

)×(1-

3

3×5

)×(1-

3

4×6

)×(1-

3

5×7

)×…×(1-

3

96×98

)×(1-

3

97×99

)．

Answer 1

分析：因为1-

3

(K-1)(K+1)

=

(K-1)(K+1)-3

(K-1)(K+1)

=

(K2-1)-3

(K-1)(K+1)

=

K2-22

(K-1)(K+1)

=

(K-2)(K+2)

(K-1)(K+1)

，所以原式=

(3-2)(3+2)

(3-1)(3+1)

×

(4-2)(4+2)

(4-1)(4+1)

×

(5-2)(5+2)

(5-1)(5-1)

×…×

(97-2)(97+2)

(97-1)(97+1)

×

(98-2)(98+2)

(98-1)(98+1)

=

1×5

2×4

×

2×6

3×5

×

3×7

4×6

×…×

95×99

96×98

×

96×100

97×99

=

1

4

×

100

97

=

25

97

．

解答：解：（1-

3

2×4

）×（1-

3

3×5

）×（1-

3

4×6

）×（1-

3

5×7

）×…×（1-

3

96×98

）×（1-

3

97×99

），
=

(3-2)(3+2)

(3-1)(3+1)

×

(4-2)(4+2)

(4-1)(4+1)

×

(5-2)(5+2)

(5-1)(5-1)

×

(6-2)(6+2)

(6-1)(6+1)

×…×

(97-2)(97+2)

(97-1)(97+1)

×

(98-2)(98+2)

(98-1)(98+1)

，
=

1×5

2×4

×

2×6

3×5

×

3×7

4×6

×

4×8

5×7

…×

95×99

96×98

×

96×100

97×99

，
=

1

4

×

100

97

=

25

97

．

点评：完成本题要细心分析式中的数据，从中找出内在联系，然后进行巧算．