A Pattern I Randomly Found

I don't quite remember how I found this, but when you put in:

$f(n) = \frac{10^{n}+8}{18}$

You get the pattern:

$1 = 1$

$2 = 6$

$3 = 56$

$4 = 556$

$5 = 5556$

etc etc

Having changed it a bit, I changed it to:

$f(n) = \frac{10^{n+1}+8}{18}-1$

And you get the pattern:

$1 = 5$

$2 = 55$

$3 = 555$

$4 = 5555$

$5 = 55555$

etc etc

I just wondered if this pattern has any significance...

#NumberTheory

Easy Math Editor

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Comments

If you modify the second $f(n)$ , you end up with

$f(n) = \frac{10^{n+1}-10}{18} = \frac{5(10^n-1)}{9}.$ .

$10^n - 1 = 9 \sum_{i=0}^{n-1} 10^i,$

you have that

$f(n) = 5 \times \sum_{i=0}^{n-1} 10^i.$

In decimal form, this represents a string of $n$ $5$ 's, which explains the phenomenon you see.

To explain the first $f(n)$ , rewrite it as

$f(n) = \frac{10^{n+1}-10}{18} + 1$

and apply the same principle as above. Long story short: what you have found is not a random pattern but rather formulas that turn out to have interesting decimal representations.

A Former Brilliant Member - 2 years, 7 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$