probably a simple answer to this

A

Avahlanch

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Sep 29, 2017
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I want a simple formula to basically take 10^10 = x, and then do x^10 =, and repeat to n amount of times. I was looking at arithmetic series', but couldnt figure out a way to write a formula that will take the sum and repeat the original equation onto it
 
First Impression: No idea what it is you are trying to do.

Second Effort
[math]If\;10^{10} = x,\;then\; x^{10} = \left(10^{10}\right)^{10}[/math], but what have we accomplished toward your goal?

Repeat what "n amount of times"?

Last Impression: "take the sum and repeat the original equation onto it" - definitely no idea what that means.

Please try again. Show some of your efforts and we can probably figure out what it is you are trying to do.
 
If what you're asking for is a simple formula for 10^(10^(10^...10))), with n 10's in this "power tower", the answer is that there is no simple formula. Rather, a simple notation can be invented representing this calculation, and used to express very large numbers that can't be expressed in other ways. See https://en.wikipedia.org/wiki/Tetration.

On the other hand, you appear to be doing this: (...((10^10)^10)^...)^10, which is in fact much simpler. Try simplifying this at each step: What is (10^10)^10?
 
For example, get a calculator, and do 1+1=. then press = again. It will repeat the previous action onto your new sum. I want to do this with exponents, and use a variable to determine how many times you press =. (and obviously not need to press = repeatedly, and yes i understand it will be ridiculously large numbers even in scientific notation.)
 
Presumably, you are assuming that each time you (figuratively) "press =", it will raise the number in the display to the 10th power, just as when you press = (in some calculators) after entering "1 + 1", it will add 1 again:

1 + 1 = 2​
= 3 [i.e. 2 + 1 = 3]​
= 4 [i.e. 3 + 1 = 4]​
...​

and the result after doing this n times (I did it 3 times) will be f(n) = n+1. That's the sort of answer you are asking for, a simple formula for the result of applying the same operation n times.

That is, the process can be expressed in terms of functions as

f(1) = 1 + 1 = 2​
f(2) = f(1) + 1 = 3​
f(3) = f(2) + 1 = 4​

Have I got that right?

So you want this:

10^10 = 1*10^10 [scientific notation]​
= 1*10^100 [i.e. (10^10)^10 = 1*10^100]​
= 1*10^1000 [i.e. ((10^10)^10)^10 = 1*10^1000]​
...​

Or, in terms of a function,

f(1) = 10^10​
f(2) = f(1)^10 = (10^10)^10​
f(3) = f(2)^10 = ((10^10)^10)^10​
...​

Does that match what you want?

Now, please give it some thought. Can you simplify (10^10)^10? Then, can you simplify ((10^10)^10)^10? Then you'll be able to find a formula. The numbers aren't nearly as large as they would be if you had meant 10^(10^(10^...10))) , as I first suggested.
 
To finish off, we have this after simplification:

f(1) = 10^10​
f(2) = f(1)^10 = (10^10)^10 = 10^(10*10) = 10^(10^2)​
f(3) = f(2)^10 = (10^(10^2))^10 = 10^(10^2*10) = 10^(10^3)​
...​
f(n) = 10^(10^n)​

That is the formula you are asking for. Each time you raise to the 10th power, you are multiplying the exponent by 10.
 
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