1) 101+1011=
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
2) 101+ 1011+ 10111 + 101111 ... ... + 101111111111111111=
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
3) 101+1101+11101+111101 ... ... +111111111101=
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
4) 101+ 110011 + 111000111 + 111100001111... ... +
111111111100000000001111111111 =
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
5) 111+222+333+444+555+666+777+888+999+ 101010 =
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
6) 123+456+ 789=
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555 =
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
8) 7+8+9+10+11+12 ... ... 1111 =
圖片參考：http://l.yimg.com/f/i/tw/ugc/rte/smiley_6.gif
101 + 1011 + 10111 ... ... ?
2009-07-02 10:11 pm
回答 (3)
2009-07-05 3:50 am
呢題唔投啦
因為個個答案案唔似最佳
都答唔晒全部既8題
因為個個答案案唔似最佳
都答唔晒全部既8題
2009-07-03 1:14 am
1) 101+1011= 1112 (straight forward)
2) 101+ 1,011+ 10,111 + 101,111 ... ... + 101,111,111,111,111,111=
First separate the number into 2 parts, higher and lower than the zero digit.
Higher part
= 100 + 1,000 + 10,000 + ... + 100,000,000,000,000,000
= 111,111,111,111,111,100
Lower part
= 1 + 11 + 111 + 1,111 + ... + 1,111,111,111,111,111
=1/9 x (9 + 99 + 999 + 9,999 + ... + 9,999,999,999,999,999)
=1/9 x (10-1 + 100-1 + 1000-1 + 10000-1 + ... + 10,000,000,000,000,000-1)
=1/9(11,111,111,111,111,111 - 16)
=1,234,567,901,234,567又7/9-16/9
=1,234,567,901,234,566
So the sum is 111,111,111,111,111,100 + 1,234,567,901,234,566
= 112,345,679,012,345,666
3) 101+1,101+11,101+111,101 ... ... +111,111,111,101
=(100 + 1,100 + 11,100 + 111,100 +...+ 111,111,111,100) + 1 x 10
=A x 100 + 10
where A = 1 + 11 + 111 + 1,111 +...+ 1,111,111,111
A = 1/9(9 + 99 + 999 + 9,999 +...+ 9,999,999,999)
=1/9(10 + 100 + 1,000 + 10,000 + ...+ 10,000,000,000 - 10)
=1/9(11,111,111,110 - 10)
=1,234,567,901又1/9 - 10/9
=1,234,567,900
So sum is A x 100 + 10
=123,456,790,010
4) 101+ 110,011 + 111,000,111 + 111,100,001,111... ... + 111,111,111,100,000,000,001,111,111,111
First separate the number into 2 parts, higher and lower than the zero digit.
Higher part
= 100 + 110,000 + 111,000,000 + 111,100,000,000 + ...+ 111,111,111,100,000,000,000,000,000,000
=1/9(9x10^2 + 99x10^4 + ... + 9999999999x10^20)
=1/9(10x10^2 - 1x10^2 + 10^2x10^4 - 1x10^4 +...+ 10^10x10^20 - 1x10^20)
=1/9(1,001,001,001,001,001,001,001,001,001,000 - 101,010,101,010,101,010,100)
=1/9(10,010,010,000,899,990,899,990,899,990,900)
=111,222,333,433,332,322,221,211,110,100
Lower part = 1 + 11 + 111 + ...+ 1,111,111,111
= 1,234,567,900 [see the value of A in (3)]
So the required sum
=111,222,333,433,332,322,221,211,110,100 + 1,234,567,900
=111,222,333,433,332,322,222,445,678,000
2009-07-02 17:15:12 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010 =
= 111 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) + 101,010
= 111 x (9 + 1) x 9 / 2 + 101,010
= 4,995 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368 (straight forward)
2009-07-02 17:15:27 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1 + 5555) x 5555 / 2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111) x (1111 - 7 + 1)/2
= 1118 x 1105 / 2
= 617,695
2009-07-02 20:27:24 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:27:34 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:28:36 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:28:50 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:31:07 補充：
為甚麼總是沒法補充內容:
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:31:18 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:32:57 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:33:08 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:35:35 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:35:48 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-03 08:59:45 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-03 09:00:03 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2) 101+ 1,011+ 10,111 + 101,111 ... ... + 101,111,111,111,111,111=
First separate the number into 2 parts, higher and lower than the zero digit.
Higher part
= 100 + 1,000 + 10,000 + ... + 100,000,000,000,000,000
= 111,111,111,111,111,100
Lower part
= 1 + 11 + 111 + 1,111 + ... + 1,111,111,111,111,111
=1/9 x (9 + 99 + 999 + 9,999 + ... + 9,999,999,999,999,999)
=1/9 x (10-1 + 100-1 + 1000-1 + 10000-1 + ... + 10,000,000,000,000,000-1)
=1/9(11,111,111,111,111,111 - 16)
=1,234,567,901,234,567又7/9-16/9
=1,234,567,901,234,566
So the sum is 111,111,111,111,111,100 + 1,234,567,901,234,566
= 112,345,679,012,345,666
3) 101+1,101+11,101+111,101 ... ... +111,111,111,101
=(100 + 1,100 + 11,100 + 111,100 +...+ 111,111,111,100) + 1 x 10
=A x 100 + 10
where A = 1 + 11 + 111 + 1,111 +...+ 1,111,111,111
A = 1/9(9 + 99 + 999 + 9,999 +...+ 9,999,999,999)
=1/9(10 + 100 + 1,000 + 10,000 + ...+ 10,000,000,000 - 10)
=1/9(11,111,111,110 - 10)
=1,234,567,901又1/9 - 10/9
=1,234,567,900
So sum is A x 100 + 10
=123,456,790,010
4) 101+ 110,011 + 111,000,111 + 111,100,001,111... ... + 111,111,111,100,000,000,001,111,111,111
First separate the number into 2 parts, higher and lower than the zero digit.
Higher part
= 100 + 110,000 + 111,000,000 + 111,100,000,000 + ...+ 111,111,111,100,000,000,000,000,000,000
=1/9(9x10^2 + 99x10^4 + ... + 9999999999x10^20)
=1/9(10x10^2 - 1x10^2 + 10^2x10^4 - 1x10^4 +...+ 10^10x10^20 - 1x10^20)
=1/9(1,001,001,001,001,001,001,001,001,001,000 - 101,010,101,010,101,010,100)
=1/9(10,010,010,000,899,990,899,990,899,990,900)
=111,222,333,433,332,322,221,211,110,100
Lower part = 1 + 11 + 111 + ...+ 1,111,111,111
= 1,234,567,900 [see the value of A in (3)]
So the required sum
=111,222,333,433,332,322,221,211,110,100 + 1,234,567,900
=111,222,333,433,332,322,222,445,678,000
2009-07-02 17:15:12 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010 =
= 111 (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) + 101,010
= 111 x (9 + 1) x 9 / 2 + 101,010
= 4,995 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368 (straight forward)
2009-07-02 17:15:27 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1 + 5555) x 5555 / 2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111) x (1111 - 7 + 1)/2
= 1118 x 1105 / 2
= 617,695
2009-07-02 20:27:24 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:27:34 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:28:36 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:28:50 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:31:07 補充：
為甚麼總是沒法補充內容:
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:31:18 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:32:57 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:33:08 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 20:35:35 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-02 20:35:48 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-03 08:59:45 補充：
5) 111+222+333+444+555+666+777+888+999+ 101,010
= 111 x (1+2+3+4+5+6+7+8+9)+101,010
= 111 x 10x9/2 + 101,010
= 106,005
6) 123 + 456 + 789 = 1,368
2009-07-03 09:00:03 補充：
7) 1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
= (1+5555)*5555/2
= 15,431,790
8) 7+8+9+10+11+12 ... ... 1111
= (7 + 1111)(1111 - 7 +1)/2
= 617695
2009-07-02 10:25 pm
Question 1
101+1011=1112
Question 5
111+222+333+444+555+666+777+888+999+ 101010
=(111+999)/2 x 9+101010
=4995+101010
=106,005
[Since 111, 222, 333,.....999 is a A. P., we can use the sum of A. P. to compute the answer of this question]
Question 6
123+456+789=1,368
Question 7
1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
=(1+5555)/2 x 5555
=15,431,790
Question 8
7+8+9+10+....+1111
=(7+1111)/2 x (1111-6)
=617,695
101+1011=1112
Question 5
111+222+333+444+555+666+777+888+999+ 101010
=(111+999)/2 x 9+101010
=4995+101010
=106,005
[Since 111, 222, 333,.....999 is a A. P., we can use the sum of A. P. to compute the answer of this question]
Question 6
123+456+789=1,368
Question 7
1+ 2+3+4+5+6+7+8+9+10+11+12+13+14+15+16+17+18+19+20... ... + 5555
=(1+5555)/2 x 5555
=15,431,790
Question 8
7+8+9+10+....+1111
=(7+1111)/2 x (1111-6)
=617,695
參考: ME
收錄日期: 2021-04-23 23:22:37
原文連結 [永久失效]:
https://hk.answers.yahoo.com/question/index?qid=20090702000051KK00935