Suppose a Square Image Has $N^{2}$ Pixels We Would Like to Approximately Know How Many Pixels

Essay

Suppose a square image has $N^{2}$ pixels. We would like to approximately know how many pixels there are in total in a $(P+1)$ -level image pyramid consisting of the original image plus $P$ smaller images, each of which is $1 / 4$ the size. As a first approximation, let's just count all possible levels, down to a size 1x1 image.
(a) First, suppose $N$ is a power of 2 .
What is an expression for the exact count of pixels in this case, if $N=2^{M}$ ?
What is the exact count of pixels in this case, if $N=16$ ? Write the total as a binary and as a decimal number.
(b) Suppose $N$ is not a power of 2 .
Just give an upper bound for the number of pixels, assuming there are an infinite number of pyramid levels and we can use floats for numbers of pixels. What is this upper bound if $N=16$ ?
Hint: for $x<1$ the Taylor series expansion of $1 /(1-x)$ is $1+x+x^{2}+x^{3}+\ldots$ . What does this mean if $x=1 / 4$ ?