Question 1208833
<pre>

I noticed that the decimal you gave 9.765625 when multiplied by 10<sup>-4</sup>
is indeed an approximation of one of the terms of the series, so maybe you left
off the " x 10<sup>-4</sup> ".  I'll assume your problem was supposed to be:  

Find the sum of this series:

{{{0.5+0.25+0.125+""*""*""*""+matrix(1,3,9.765625,"x",10^(-4))}}}

using the formula {{{S[n]}}}{{{""=""}}}{{{a[1]((1-r^n)/(1-r^""))}}}

We know this is a geometric series with {{{a[n]=matrix(1,3,9.765625,"x",10^(-4))}}}
{{{matrix(1,2,common,ratio)}}}{{{""=""}}}{{{r}}}{{{""=""}}}{{{matrix(1,2,2nd,term)/matrix(1,2,1st,term)}}}{{{""=""}}}{{{0.25/0.5=0.5}}}{{{""=""}}}{{{matrix(1,2,3rd,term)/matrix(1,2,2nd,term)}}}{{{""=""}}}{{{0.125/0.25}}}{{{""=""}}}{{{0.5}}}

But first we must find n, the number of terms, using the formula for the
nth term:

{{{a[n]}}}{{{""=""}}}{{{a[1]r^(n-1)}}}

{{{matrix(1,3,9.765625,"x",10^(-4))}}}{{{""=""}}}{{{0.5*(0.5)^(n-1)}}}

{{{matrix(1,3,9.765625,"x",10^(-4))}}}{{{""=""}}}{{{(0.5)^n}}}

We solve for n by taking the log base 10 of both sides

{{{log((9.765625))+log((10^(-4)))}}}{{{""=""}}}{{{log((0.5^n))}}}

{{{log((9.765625))-4)}}}{{{""=""}}}{{{n*log((0.5))}}}

{{{0.9897000434-4}}}{{{""=""}}}{{{n*(-0.3010299957)}}}

{{{-3.010299957/(-0.3010299957)}}}{{{""=""}}}{{{n}}}

{{{9.999999999}}}{{{""=""}}}{{{n}}}

We round that to 10

{{{10}}}{{{""=""}}}{{{n}}}

So there are n=10 terms in the series:

Using the formula:

{{{S[n]}}}{{{""=""}}}{{{a[1]((1-r^n)/(1-r^""))}}}

{{{S[10]}}}{{{""=""}}}{{{0.5((1-0.5^10)/(1-0.5^""))}}}

{{{S[10]}}}{{{""=""}}}{{{0.5((0.9990234375)/0.5))}}}

{{{S[10]}}}{{{""=""}}}{{{0.9990234375}}} <---ANSWER

So you can see that the sum of the series to 10 terms is 
getting close to 1.

Edwin</pre>