# A geometric sequence using one digit

Give the first few terms of a geometric[1] sequence such that:

• the sequence is increasing and infinite
• only one digit is used throughout the sequence
• the terms are written as base ten decimals and are integers

[1]For a geometric sequence, each subsequent term is found by multiplying the previous one by a fixed non-zero number.

• Do you have a particular sequence in mind that you know exists? – AHKieran Dec 20 '19 at 9:37
• @AHKieran, yes, one, and replies may be with its first few terms, instead of later terms of the same seq. – Tom Dec 20 '19 at 9:41
• Are you asking for the first few terms because the pattern of only one digit breaks at some point? Because a geometric sequence is uniquely determined by its first term and its common ratio. – Taladris Dec 21 '19 at 7:16

## 2 Answers

How about this sequence

$$9.999\ldots$$
$$99.999\ldots$$
$$999.999\ldots$$
$$9999.999\ldots$$

as each term is

an integer equal to $$10, 100, 1000, 10000$$ etc

• appears to fail in the requirement that the terms are integers – Penguino Dec 20 '19 at 10:04
• @Penguino these are integers – hexomino Dec 20 '19 at 10:04
• Nice. Is that lateral-thinking? – Jay Dec 20 '19 at 11:07
• @MatthewWells This isn't rounding, those are real equal signs. – Thomas Markov Dec 20 '19 at 12:47
• Are you using 10 as the fixed non-zero integer? – Smock Dec 20 '19 at 13:52

There are infinite of these sequences

All have their geometric factor (not sure about this term) of

1x

Example:

0, 0, 0, ...
5, 5, 5, ...
11, 11, 11, ...
888, 888, 888, ...
...

• Maybe this is strictly correct but I meant strictly increasing :P – Tom Dec 20 '19 at 9:20
• The "geometric factor" is usually called the common ratio of the sequence, since it is the ratio that any two consecutive terms share. – Taladris Dec 21 '19 at 7:25