整數(shù)裂項(xiàng)

娜喃撥

整數(shù)裂項(xiàng)基本公式

(1)lx2+2x3+3x4+...+(n-l)XM=-(?-l)xnx(n+l)

(2)1x2x3+2x3x4+3x4x5+...+(n-2)x(?-l)xn=—(n-2)(〃-\)n{n+1)

4

[例1]Ix2+2x3+3x4+…+49x50=

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】這是整數(shù)的裂項(xiàng)。裂項(xiàng)思想是：瞻前顧后，相互抵消。

設(shè)S=Ix2+2x3+3x4+…+49x50

1x2x3=1x2x3

2x3x3=2x3x(4-1)=2x3x4-1x2x3

3x4x3=3x4x(5-2)=3x4x5—2x3x4...

49x50x3=49x50x(51-48)=49x50x51-48x49x50

3S=1x2x3+2x3x3+3x4x3+…+49x50x3=49x50x51

5=49x50x51^3=41650

【答案】41650

【鞏固】1x2+2x3+3x4+4x5+5x6+6x7+7x8+8x9+9x10=

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】本題項(xiàng)數(shù)較少，可以直接將每一項(xiàng)乘積都計(jì)算出來再計(jì)算它們的和，但是對(duì)于項(xiàng)數(shù)較多的情況顯然

不能這樣進(jìn)行計(jì)算.對(duì)于項(xiàng)數(shù)較多的情況，可以進(jìn)行如下變形：

/、n(n+l)(n+2)-(n-l)n(n+l)1.、/.1,、/、

"(〃+1)=---------------------------=—?(?+1)(?+2)--(n-1)/?(/!+1),

^TH4^A=—X1X2X3+|—x2x3x4--xlx2x3|+???+[—x9xl0xll--x8x9xl0|

3(33J(33)

=-x9x10xll=330

3

另解：由于〃(〃+l)=/+〃，所以

原式=02+])+Q2+2)+…+(9?+9)

=(12+22+---+92)+(1+2+---+9)=1X9X10x19+^x9x10=330

采用此種方法也可以得到1X2+2X3H+=++這一結(jié)論.

【答案】330

[例2]1x4+4x7+7x10+…+49x52=

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】設(shè)5=1x4+4x7+7x10+…+49x52

1x4x9=1x4x7+1x4x2

4x7x9=4x7x(10—1)=4x7x10—1x4x7

7xl0x9=7xl0x(13-4)=7x10x13-4x7x10

49x52x9=49x52x(55-46)=49x52x55-46x49x52

95=49x52x55+1x4x2

S=(49x52x55+1x4x2)+9=15572

【答案】15572

[例3]1X2X3+2X3X4+3X4X5H----1-9x10x11=

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】"("+1)(〃+2)=；"(〃+1)("+2)(〃+3)—；("-1)"("+1乂〃+2),所以，

=—xlx2x3x4+|—x2x3x4x5--xlx2x3x4|+---+|—x9xl0xllxl2-—x8x9xl0xll

4(444

=-x9x10xllxl2=2970

4

從中還可以看出，1x2x3+2x3x4+3x4x5+…+〃x(〃+1)x(〃+2)=；〃(〃+1)(〃+2)(〃+3)

【答案】2970

［例4］計(jì)算:Ix3x5+3x5x7+...+17x19x21=.

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】可以進(jìn)行整數(shù)裂項(xiàng).

___3x5x7x9-lx3x5x7

3x5x7=---------------------------

8

5x7x9x11-3x5x7x9

5x7x9=

……17x19x21x23-15x17x19x21

17x19x21=--------------------------------------,

8

所以原式

,cu3x5x7x9-lx3x5x717x19x21x23-15x17x19x21

=1x3x5+---------------------------+…+---------------------------------------

88

,cu17x19x21x23-1x3x5x717x19x21x23+1x3x5

=1x3x5+---------------------------------=-----------------------------=19503

88

也可適用公式.

原式=(3—2)x3x(3+2)+(5-2)x5x(5+2?.?+(19—2)xl9x(19+2)

=(32-22)X3+(52-22)X5+---+(192-22)X19

=(33+53+--.+19,)-4X(3+5+---+19)

=(P+3，+53+...+193)—4x(1+3+5+…+19)+3

而P+33+53+...+193=(F+23+33+...+203)-Q3+43+63+...+203)

=-x2O2x212-8xlxlO2xll2=19900,

44

1+3+5+…+19=1()2=100,所以原式=19900-4x100+3=19503.

【答案】19503

【鞏固】計(jì)算：10x16x22+16x22x28+...+70x76x82+76x82x88

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】可進(jìn)行整數(shù)裂項(xiàng)：

原式10x16x22x28-4x10x16x22]+(16x22x28x34-10x16x22x28]

—I24)+124)

(70x76x82x88-64x70x76x82](76x82x88x94-70x76x82x88

I24)+[24

10x16x22x284x10x16x2216x22x28x3410x16x22x28

------------------------------1-------------------------------卜,?

24242424

70x76x82x88_64x70x76x8276x82x88x94_70x76x82x88

~^2A24242A~

_76x82x88x944x10x16x22

2424

76x82x88x94-4x10x16x22

24

=2147376

【答案】2147376

【鞏固】計(jì)算：Ix2x3x4+3x4x5x6+5x6x7x8+…+97x98x99x100=

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】一般的整數(shù)裂項(xiàng)各項(xiàng)之間都是連續(xù)的，本題中各項(xiàng)之間是斷開的，為此可以將中間缺少的項(xiàng)補(bǔ)上,

再進(jìn)行計(jì)算.

亍己原式為A,再設(shè)3=2x3x4x5+4x5x6x7+6x7x8x9+--+96x97x98x99,

則A+B=lx2x3x4+2x3x4x5+3x4x5x6+…+97x98x99x100

=1x97x98x99x100x101=1901009880,

現(xiàn)在知道A與3的和了，如果能再求出A與3的差，那么4、3的值就都可以求出來了.

/4-B=lx2x3x4—2x3x4x5+3x4x5x6-4x5x6x7+5x6x7X8H---1-97x98x99x100

=4x(1x2x3+3x4x5+5x6x74-...+97x98x99)

=4X[2X(22-1)+4X(42-1)+6X(62-1)+---+98X(982-1)]

=4x(2'+4,+6^+…+98，)-4x(2+4+6+…+98)

=4x8x-Lx492x502-4x1x100x49=48010200

42

所以，4=(1901009880+48010200)+2=974510040.

【答案】974510040

[例5]2004x2003-2003x2002+2002x2001-2001x2000+..-+2x1

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】3星【題型】計(jì)算

【解析】原式=2003x2+2001x2+…+3x2+lx2

=2x(1+3+5+???+2001+2003)

=2x(1+2003)x1002-2

=2008008

其中也可以直接根據(jù)公式1+3+5+7+―+(2〃-1)="2得出

1+3+5+…+2001+2003=10022

【答案】2008008

【例6]lxl!+2x2!+3x3!+---+2008x2008!=

【考點(diǎn)】整數(shù)裂項(xiàng)【難度】4星【題型】計(jì)算

【解析】觀察發(fā)現(xiàn)2x2!=2x2xl=(3-l)x2xl=3!-2!,

3x3!=3x3x2xl=(4—I)x3x2xl=4