101 + 1011 + 10111 ... ... ?

呢題唔投啦

因為個個答案案唔似最佳

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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

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