# Symmetry and floor function

We all know symmetrical sequences. could be fascinating at times. we will be reading 'bout symmetrical formulas.

## symmetries

a really common symmetry is in the question below

we can clearly see that each number is going from an ascending order to a nu mber and then descends until 1. $1234321$ starts in natural order{1,2,3] and reaches 4. than it goes in the reverse order. we can tell the answer is $123454321$. but we dont have a formula to prove it, or do we!?????## find the next next number in the following sequence

$1,121,12321,1234321$

## the formula and the proof

from the question above, we get that
$0^2=0, 1^2=1, 11^2= 121, 111^2= 12321, 1111^2 =1234321$
and so on.... notice that

⌊$\frac{10}{9}$⌋=11

⌊$\frac{100}{9}$⌋=111

⌊$\frac{1000}{9}$⌋=1111.

we get that ⌊$\frac{10^N}{9}$⌋=n-digited-111.....

so since n-digited $1111....^2$ makes the Nth term in the sequence,we conclude the formula:

Nth-term=$⌊\frac{10^{N}}{9}⌋^{2}$

## the tricky one

the previous one didn't really need much of a formula to determine the next term. but this will

well, the answer guessed by you is probably 12345678. but it is $12345679$ it has not much discussion, but it cant be called a symmetry, since numbers are only ascending.. well try it and the formula is## find the next next number in the following sequence

$1,12,123,1234,12345,123456,1234567$

Nth-term=$⌊\frac{10^{N}}{81}⌋$so the 9th term is$\boxed{12345679}$

## similar symmetries

well i have shown you the basics , now try makin' a formula for the following

## find the formula for the Nth term

a)9,1089,110889,11108889.... b)36,4356,443556,44435556...

**Cite as:**Symmetry and floor function.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/symmetry-and-floor-function/