Question 30035
The number of grains on each checkerboard is,
1 2 4 8 16 32 64 ...
or, in exponential notation,
2^0 2^1 2^2 2^3 2^4 2^5 2^6 ...

So the number of grains on the nth square is,
Un = 2^(n-1)
============

a) on the 24th square, U24 = 2^(24-1) = 2^23
Ans:2^23
========

b) this is a geometric progression.
The sum of the terms in a GP is given by
Sn = u0(r^n - 1)/(r - 1)
where u0 is the first term in the series and r is the common ratio.
here u0 = 1 and r = 2 and n is the number of terms in the series.
so,
Sn = 1(2^n - 1)/(2 - 1)
Sn = 2^n - 1
================

when n = 24 squares,
S24 = 2^24 - 1
==============