Question

Find the sum of the geometric sequence 1, 1/4, 1/16, 1/64, 1/256 show all work please

Answer 1

S is our sum, x is 1/4 in this case, and we're adding them up to some power of 1/4. It looks like n=4 because (1/4)^4=1/256.

\[\Large S= x^0+x^1+x^2+...+x^n \\ \Large S-x^0= x^1+x^2+...+x^n \\ \Large \frac{S-x^0}{x^1}= x^0+x^1+...+x^{n-1} \\ \Large \frac{S-x^0}{x^1}+x^n= x^0+x^1+...+x^{n-1}+ x^n \\ \Large \frac{S-x^0}{x^1}+x^n=S\]

Now we can solve for S. \[\Large \frac{S-1}{x}+x^n=S\]

Answer 2

Thank You this helps a lot!

Answer 3

You might find this formula useful:
\[S _{n}=\frac{a(1-r ^{n})}{1-r}\]
which is the sum of n terms of a geometric series with first term a and common ratio r.