A geometric sequence using one digit

Question

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, Commented Dec 20, 2019 at 9:37
@AHKieran, yes, one, and replies may be with its first few terms, instead of later terms of the same seq. — Tom, Commented Dec 20, 2019 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, Commented Dec 21, 2019 at 7:16

hexomino · Accepted Answer · 2019-12-20 10:03:33Z

22

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

answered Dec 20, 2019 at 9:57

hexomino

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1

$\begingroup$ appears to fail in the requirement that the terms are integers $\endgroup$
– Penguino
Commented Dec 20, 2019 at 10:04
9

$\begingroup$ @Penguino these are integers $\endgroup$
– hexomino
Commented Dec 20, 2019 at 10:04
3

$\begingroup$ Nice. Is that lateral-thinking? $\endgroup$
– Jay
Commented Dec 20, 2019 at 11:07
8

$\begingroup$ @MatthewWells This isn't rounding, those are real equal signs. $\endgroup$
– Thomas Markov
Commented Dec 20, 2019 at 12:47
1

$\begingroup$ Are you using 10 as the fixed non-zero integer? $\endgroup$
– Smock
Commented Dec 20, 2019 at 13:52

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Omega Krypton · Accepted Answer · 2019-12-20 09:20:57Z

3

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, ...
...

answered Dec 20, 2019 at 9:17

Omega Krypton

21.7k2 gold badges25 silver badges125 bronze badges

5

$\begingroup$ Maybe this is strictly correct but I meant strictly increasing :P $\endgroup$
– Tom
Commented Dec 20, 2019 at 9:20
$\begingroup$ The "geometric factor" is usually called the common ratio of the sequence, since it is the ratio that any two consecutive terms share. $\endgroup$
– Taladris
Commented Dec 21, 2019 at 7:25

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