chess "squares"

Hi Nicko,

Is there any other way of finding the answer for this except for doing the operation?
2^0+2^1+2^2+…+2^63

This is the sum of a geometric progression GP for short.
Each number in a number pattern can be called a term, So you have a 1st term, a second term etc.

These are examples of GPs
5,50,500,5000 because each term is 10* the term before it
120,60,30,15.7,5, because each term is 0.5 * the term before it
Any number pattern where the next number is found by multiply the one before by a specific number (r) is called a Geometric progression
r is that specific number it is called the common ratio
because
r = term3/term2 = term5/term4 = any term/the term before it.

What is r in your series ? That is, what is each term multiplied by to get the next term?
You also need n.
n is the number of terms that you want to add together, Be careful, the answer is not 63, can you see why not?
a, also called T ₁ is the first term.

The sum of a GP is given by the following formula

Sn = a(r ⁿ-1) / (r-1)

When you work out what a,r and n are then you can substitute them in and work out what the answer is.
It is a very big number so your calculator will probably give the answer in scientific notation
If you use the web2.0calc, it will give you the answer in normal notation

NickOs:

Is there any other way of finding the answer for this except for doing the operation?
2^0+2^1+2^2+…+2^63

2^0= 1
2^0+2^1= 3
2^0+2^1+2^2= 7
2^0+2^1+2^2+2^3= 15
...

1=2^1-1
3=2^2-1
7=2^3-1
15=2^4-1
...
2^0+2^1+2^2+…+2^63= 2^64-1