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

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Chill8160

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May 2, 2018
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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!


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Chill8160 said:
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
Click to expand...
When you say START with - do you mean the unit digit is "1" - like "9^2 = 81" or like "9^17 =16677181699666569"
 
Subhotosh Khan said:
When you say START with - do you mean the unit digit is "1" - like "9^2 = 81" or like "9^17 =16677181699666569"


16677181699666569

Click to expand...

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!!
 
Subhotosh Khan said:
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!!
Click to expand...

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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