(1)①63+73+83+93+103+113+123+133+143+153 ②(1+
③(
(2)在自然数1~60中找出8个不同的数,使这8个数的倒数之和等于1. (3)
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(1)①63+73+83+93+103+113+123+133+143+153,
=(1+2+3+…+15)2-(1+2+3+…+5)2,
=14400-225,
=14175;
②设1+
+1 2
+1 3
=a,1+1 4
+1 2
+1 3
+1 4
=b,1 5
(1+
+1 2
+1 3
)×(1 4
+1 2
+1 3
+1 4
)-(1+1 5
+1 2
+1 3
+1 4
)×(1 5
+1 2
+1 3
),1 4
=a×(b-1)-b(a-1),
=ab-a-ab+b,
=b-a,
=1+
+1 2
+1 3
+1 4
-(1+1 5
+1 2
+1 3
),1 4
=
;1 5
③(
+1 2
+1 3
+…+1 4
)+(1 20
+2 3
+2 4
+…+2 5
)+(2 20
+3 4
+…+3 5
)+…+(3 20
+18 19
)+18 20
,19 20
=
+(1 2
+1 3
)+(2 3
+1 4
+2 4
)+…+(3 4
+1 20
+…+2 20
+18 20
),19 20
=(1+2+3+4+5+6+7+8+9)+
+1 2
+3 2
+5 2
+7 2
+9 2
+11 2
+13 2
+15 2
+17 2
,19 2
=45+50,
=95;
(2)因为1=
+1 1×2
+1 2×3
+1 3×4
+1 4×5
+1 5×6
+1 6×7
+1 7×8
,1 8
=
+1 2
+1 6
+1 12
+1 20
+1 42
+1 56
,1 8
所以这8个不同的数是2、6、8、12、20、30、42、56.
(3)
-3 2
+5 6
-7 12
+9 20
-11 30
+13 42
,15 56
=(1+
)-(1 2
+1 2
)+(1 3
+1 3
)-(1 4
+1 4
)+(1 5
+1 5
)-(1 6
+1 6
)+(1 7
+1 7
),1 8
=1+
,1 8
=
.9 8