Question 927192
This is about {{{exponential}}}{{{ relationships}}}, in which a quantity grows larger or smaller at an {{{increasing}}}{{{ rate}}} rather than at a {{{constant}}}{{{ rate}}} in linear relationships.

Exponential growth is everywhere in the world. It isn't just counting how many Rubas a peasant gets from the king in a math book. One example of exponential growth in real life is allowance. Say your daily allowance is {{{25}}} cents. You make a deal with your parents to {{{double}}} your allowance daily for {{{10}}} days. It wouldn't seem like you were making much money at the start, but at the end of the {{{10}}} days, you would have made ${{{128}}}!

When I think about {{{Real}}} life use for {{{Exponential}}} Relationships, I think about {{{money}}}. Taxes involve Exponential Relationships too. They have a specific pattern to them and if you follow it correctly you'll know how much your next bill will be. :)

here is the story about rubas:

One day in the ancient kingdom of Montarek, a peasant saved the life of the king’s daughter.  The king was so grateful he told the peasant she could have any reward she desired.  
The peasant—who was also the kingdom’s chess  champion—made an unusual request:

“I would like you to place {{{1}}} ruba on the first square of my chessboard, {{{2}}} rubas on the second square, {{{4}}} on the third square, {{{8}}} on the fourth square, and so on, until you have covered all {{{64}}}} squares. Each square should have {{{twice}}} as many rubas as the {{{previous}}} square.”

The king replied, “Rubas are the least valuable coin in the kingdom. Surely you can think of a better reward.” But the peasant insisted, so the king agreed to her request.

Did the peasant make a wise choice?

think about this request:
{{{1}}} ruba on the first square of my chessboard, 
{{{2}}} rubas on the second square, 
{{{4=2^2}}} on the third square, 
{{{8=2^3}}}
{{{16=2^5}}}
{{{32=2^6}}}
{{{64=2^8}}}
{{{128=2^9}}}
{{{256=2^10}}}..........{{{9th}}} square
{{{512=2^11}}}
:
:
and so on,  up to {{{64th}}} square 

an equation for the relationship between the number of the square n and the number of rubas, {{{r}}} is:

{{{r=2^(n-1)}}} or {{{r=(1/2)*2^n }}}

Exponential vs Linear growth :-) life lesson: always try to choose better deal