Question 1085040
this is a geometric series.


the formula for sum of the surface area becomes Sn = 64 * pi * (1 - (1/4)^n) / (1 - (1/4))


the formula for the sum of the volume becomes Sn = (256/3) * pi * (1 - (1/16)^n) / (1 - 1/16)


the formula for each succeeding surface area becomes An = 64 * pi * (1/4)^(n-1)


the formula for each succeeding volume becomes An = (256/3) * (1/16)^(n-1)


i tested these formulas out using excel and they agree with what each of the values should be.


you can test these formulas out for n = 1 through 4 and you'll see that they work.


my excel worksheet for doing it manually through finding the number of spheres and the radius of each of the spheres is sh own below.


<img src = "http://theo.x10hosting.com/2017/061906.jpg" alt="$$$" </>


my excel worksheet for doing it via the formulas is shown below:


<img src = "http://theo.x10hosting.com/2017/061907.jpg" alt="$$$" </>


the numbers match.


if my assumptions were right about what you were looking for, then the formulas are right.


i don't have time to go through the derivation of the formulas right now, but if you want or need to know, then send me an email and i'll work them up for you.


notice the excel worksheet gives the results as being divided by pi.


this means the actual result is what is shown * pi.


therefore, if you see 16 as a result, it's really 16 * pi, etc.