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1、1234561 2 3 4 5 6赋值法求二项式系数和例 1.若(1-2 x) 6 =a +a x +a x 2 +a x 3 +a x 4 +a x 5 +a x0 1 2 3 4 5 66,则(1)a +a +a +a +a +a +a = 0 1 2 3 4 5 6_;(2)a +a +a +a = 0 2 4 6_;(3)a +a +a = 1 3 5_;(4)| a | +| a | +| a | +| a | +| a | +| a | +| a |= 0 1 2 3 4 5 6_;(5)( a +a +a +a ) 0 2 4 62-(a +a +a ) 1 3 52=_;(6)
2、a +2 a +2 0 12a +223a +234a +245a +256a =6_;(7)a -2 a +2 2 a -23 a +2 4 a -25 a +2 6 a = 0 1 2 3 4 5 6_;(8)(9)a +2 a +3a +4 a +5a +6 a = 1 2 3 4 5 6a -2 a +3a -4 a +5a -6 a = 1 2 3 4 5 6_;_;(10)a +01 1 1 1 1 1 a + a + a + a + a + a =2 3 4 5 6 7_;(11)-a +01 1 1 1 1 1 a - a + a - a + a - a =2 3 4 5 6
3、7_.解析:(1)令x =1,则a +a +a +a +a +a +a =1 0 1 2 3 4 5 6;(2)令x =-1,则 a -a +a -a +a -a +a =360 1 2 3 4 5 6=729,故a +a +a +a = 0 2 4 61 +7291=365;(3)由(1)(2)知a +a +a = 1 3 51 -7291=-364;234551234560123456123 4 5 61 2 3 4 5 6(4)| a | +| a | +| a | +| a | +| a | +| a | +| a |=a -a +a -a +a -a +a =729 0 1 2 3
4、4 5 6 0 1 2 3 4 5 6;(5)平方差公式,( a +a +a +a ) 2 -(a +a +a ) 2 =( a +a +a +a +a +a +a )(a -a +a -a +a -a +a ) 0 2 4 6 1 3 5 0 1 2 3 4 5 6 0 1 2 3 4 5 6=1 36 =729(6)令x =2,则a +2a +2 0 12a +232a +234a +254a +256a =(1-4) 66=729;(7)令x =-2,则a -2a +2 2 a -23 a +2 4 a -25 a +26 a =(1+4) 6 =56 =15625 0 1 2 3 4
5、5 6;(8)对原式求导,a +2a x +3a x 1 2 32+4a x43+5a x54+6a x65=6(1-2 x)5( -2) ,令 x =1 ,则a +2a +3a +4a +5a +6a =6(1-2) 1 2 3 4 5 65( -2) =12;(9)对原式求导,a +2 a x +3a x +4a x +5a x +6a x =6(1-2 x ) (-2)1 2 3 4 5 6a -2a +3a -4a +5a -6a =6(1+2) 5 ( -2) =-2916则;1 2 3 4 5 6,令x =-1,( 10 )原式可由a x +01 1 1 1 1 1 1a x 2
6、+ a x 3 + a x 4 + a x 5 + a x 6 + a x 7 =- (1-2 x) 2 3 4 5 6 7 147求导得到,令x =1,则1 1 1 1 1 1 1 1 a + a + a + a + a + a + a =- (1-2) 7 =2 3 4 5 6 7 14 14;( 11 )原式可由a x +01 1 1 1 1 1 1a x2 + a x3 + a x4 + a x5 + a x6 + a x7 =- (1 -2 )x7 2 3 4 5 6 7 14求导得到,令x =-1,则-a +01 1 1 1 1 1 1 3 a - a + a - a + a - a =- + (1 =2-)7 2 3 4 5 6 7 14 147=-218714.