What is the generic term of the sequence if the numbers are 1, 1 000, 10 000 000, 1 000 000 000 000 000? And why?

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embizze's profile pic

embizze | High School Teacher | (Level 1) Educator Emeritus

Posted on

We are given the sequence 1, 1000, 10000000, 1000000000000000

First,  there are an infinite number of sequences that start with any given finite number of terms. We assume that we are looking for the "obvious" sequence, but any sequence that you can justify would be allright.

Consider the number of zeros (or equivalently the power of 10): these form a sequence 0,3,7,15,... 

I assume there is a typo in the problem as (1) I cannot find this as an "obvious seuence"; OEIS (the online encyclopedia of integer sequences) does not have a match and it is not one of the normal sequences and (2) it is very close to a typical sequence that is in OEIS.

Assuming the sequence is 10, 1000, 10000000, 1000000000000000, etc... or 1, 10, 1000, ... the powers are:

0,1,3,7,15,... and the nth term is 2^(n-1)-1. (In the case of 10,1000,... the nth term is 2^(n)-1)

If the sequence is 10, 1000, 10000000,... the nth term is 10^(2^(n)-1).

If the sequence is 1, 10, 1000, 10000000,... the nth term is 10^(2^(n-1)-1)

Sources:
nick-teal's profile pic

nick-teal | High School Teacher | (Level 3) Adjunct Educator

Posted on

Are you sure the 3rd term is 10,000,000 instead of 100,000,000?

If it was 100,000,000 then the series would be 

`A_n = 10^(n^2-1)`

I am having a hard time coming up for a series with 10 million instead of 100 million.

The way I found this current one is by counting the zeros.  Which are 0, 3, 8, and 15.  I noticed these are all 1 less than square numbers, which led me to this result.

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