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Find \(1_6 + 2_6 + 3_6 + \cdots + 45_6\). Express your answer in base \(6\).

 

Thanks in advance!

 Feb 16, 2019
 #1
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This problem is simply 1+2+3+4+5+6+7...

45 base 6 is 5*6^0+4*6^1 which is 29.

So 1+2+3+4+5+6+...29.

Using the finite arithmetic sequence sum n(a1+an)/2

29(1+29)/2

29*30/2

29*15

435

 Feb 17, 2019
 #2
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That is correct in base 10: You can list all the numbers in base 6 as follows:

(1, 2, 3, 4, 5, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 24, 25, 30, 31, 32, 33, 34, 35, 40, 41, 42, 43, 44, 45)=29 numbers.

Then sum them in base 10 as shown above =435 in base 10. 

Convert 435 from base 10 to base 6:

435 / 6 = 3 as the remainder.

72 / 6   = 0 as the remainder.

12 / 6   = 0 as the remainder.

2/6       = 2 as the remainder.

so 435 in base 10 =2003(from the bottom up) in base 6, which is the answer to your question.

 Feb 17, 2019

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