Is this true? (exponential math help!)?

Question

If you place a penny on a square of a chess board, and double it on every square (1 penny, 2 pennys, 4 pennies, 8 pennies etc...) by the time you reach the 64th square what would be the ammount of money? Does anyone know this? thanks!

Answer 1

A simple nth term formula -A.K.A a geometric progression- can be used.
The formula for any term is a GP= Ar^(n-1)
A=first term
r= common ratio
n= term you want to find

For example, on the 64th square there are:
1x(2^63)=9.223 x 10^17 (thats a standard form number, which means its pretty dam high, ill explain later)

The formula for the SUM of terms is:

[ A(1- R^n) ] / (1 - R)

Therefore the number of pennies on the board =

[1x (1- 2^64) ] / (-1) = 1.844674407 E 19

1.844674407 E 19 is a standard form number because its so large - this is how it showed up on my clalculator.

As a natural number, this means there are this many pennies:

18446744070000000000p

or (if your american) $184,467,440,700,000,000 - divided by 100 to get dollars

that is 3074457346 times as much money as there is is circulating in the USA (Divided it by 6trillion) -, so I wouldn't try it.... :P

Ironically I've spent all this time answering this question to avoid maths revision...

This sounds like an apporpirate answer, because think about the last sqaure alone: 2^64 = 2 x 2 x 2 x 2 x 2 x 2 ... 64 times!

Answer 2

18446744079709551615
or (2^64) - 1

---
Say you have only 4 squares.
In square 1 you have 1 penny
In square 2 you have 2 pennies
In square 3 you have 4 pennies
In square 4 you have 8 pennies
8+4+2+1 = 15
2^4 = 16 - 1 = 15

Notice the flawed logic of the other answers.
2^63 or 2^(64-1) will give you the number of pennies on the last square. Just the same, 2^3 or 2^(4-1) will give you 8, the number of pennies on the 4th square from the example above. If you want the total, just take 2 and raise it to the power of the number of squares and take 1 away from the final answer.

It's binary. The next number in a binary sequence is always 1 more than the sum of the sequence so far.
Example. 1,2,4,8,16 - the next number in the sequence is 32 (1+2+4+8+16 = 31).

Answer 3

You have a geometric series 1 + r+r^2+r^3+...+r^n.
The sum of this series is (1-r^n)/(1-r).
In your case, r = 2 and n = 63 (n goes from 0 to 63 for 64 different squares). Plug those values into the above equation for the sum and get 2^63.

So answer is 2^63/100 dollars.

Answer 4

Because you are doubling it on every square your formula is 2^n

So in this case start with=2^0=1 penny (initial location), 1rst square=2^1=2 pennies, 2nd square=2^2=4 pennies, 3rd square=2^3=8 pennies,...64th square=2^64=18,446,744,073,709,552,000 pennies

Answer 5

Money on the 64th square alone is:
Amount = 2^(63) pennies =
9,223,372,036,854,775,808 pennies.

that's about
92,234 Trillions ( a lot of money)

Just for comparison
Total National (US) debt is about
48 Trillions.

Answer 6

Square 1 corresponds to 2^0=1
Square 2 corresponds to 2^1=2
Square 3 corresponds to 2^2=4
...
...
...
Square 64 corresponds to 2^(63)

So if you want to know how much money is in the last square, that's it. If you want to know the total amount of money, then sum this geometric series to get (2^(64)-1) / (2-1) = 2^(64)-1. These numbers are in cents, of course.

Answer 7

A great place to answer this question is in Excel(r)!
Start with square A1:
put the number 1 is that square.
in square A2 put the command:
= (2*A1)
grab the bottom right corner of a2 and drag down to a64.
This should fill the numbers of each value.
The last number should be :
92,233,720,368,547,800.00
(which you might have to change the format to read)
which is equal to 2^63.

Answer 8

There is one penny on the first square, and twice as many on the second. This patern continues for 64 squares.

First square
2^0 = 1

Second square
2^1 = 2

64th square
2^63 = 2.223372037 * 10^18

In most cases I would answer 2^63 it shows you recognize the pattern.

Answer 9

you may simplify stuff by skill of taking e^x = y. After that, the equation reduces to: y^2 - 3y - 4 = 0 which, on fixing yields y = 4, -one million So, x = ln(4), ln(-one million) once you're looking only for real strategies, then the respond is x = ln(4).

Answer 10

2^64
= 18446744073709551616 pennies