The Set of 9^n Challenge: How many numbers in the set START with "9"?

Chill8160

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May 2, 2018
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5
Hello,

I was presented with this problem as a challenge and I have been trying for days and have not been able to figure it out!

The question is as follows:

There exists a set S that contains {9^0, 9^1, ... 9^4000}. In other words {9^n} for n=0 to n=4000 where n is an integer.

How many numbers in the set START with the number 9?

Any help would be EXTREMELY appreciated!

Thanks!


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Hello,

I was presented with this problem as a challenge and I have been trying for days and have not been able to figure it out!

The question is as follows:

There exists a set S that contains {9^0, 9^1, ... 9^4000}. In other words {9^n} for n=0 to n=4000 where n is an integer.

How many numbers in the set START with the number 9?

Any help would be EXTREMELY appreciated!

Thanks!


Sent from my iPhone using Tapatalk
When you say START with - do you mean the unit digit is "1" - like "9^2 = 81" or like "9^17 =16677181699666569"
 
When you say START with - do you mean the unit digit is "1" - like "9^2 = 81" or like "9^17 =16677181699666569"


16677181699666569


No. For example... the number 9, 90, 900 etc.

I know that 90 and 900 are not in the set; however, I’m using them as example.


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9^(22*1), 9^(22*2), 9^(22*3), ...... 9^(22*6) - all start with '9'

After that the pattern is broken ..... I do not know why!!
 
9^(22*1), 9^(22*2), 9^(22*3), ...... 9^(22*6) - all start with '9'

After that the pattern is broken ..... I do not know why!!

That’s what I found as well. I figure the only way to solve this is by finding the pattern but I can not seem to figure it out!

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The first few digits of 9^22 are 984...
The first few digits of 9^44 are 969...
The first few digits of 9^66 are 955...
...
The first few digits of 9^132 are 912...

That's why the pattern ends. It cycles. After 9^(22*6), the next numbers are 9^(22*7 - 1), 9^(22*8 - 1), 9^(22*9 - 1).
So on until 9^(22*14 - 2)
So on until 9^(22*21 - 3)
So on until 9^(22*28 - 4)

Extrapolate until the sequence beginning 9^(22*181 - 26)
 
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