Question 1017122
{{{8191.5}}}{{{""=""}}}{{{(expr(3/2)(4^n-1))/3}}} 
 <pre>
Write the denominator 3 as 4-1

{{{8191.5}}}{{{""=""}}}{{{expr(3/2)(4^n-1)/(4-1))}}}

Compare to the sum formula for a geometric series:

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

{{{a[1]}}}{{{""=""}}}{{{3/2}}}, {{{r}}}{{{""=""}}}{{{4}}}

Also solve for n

{{{8191.5}}}{{{""=""}}}{{{expr(3/2)(4^n-1)/3)}}}

Multiply both sides by 3

{{{8191*3}}}{{{""=""}}}{{{expr(3/2)(4^n-1)}}}

Multiply both sides by 2

{{{8191.5*3*2}}}{{{""=""}}}{{{expr(3)(4^n-1)}}}

Divide both sides by 3

{{{8191.5*2}}}{{{""=""}}}{{{4^n-1}}}

{{{16383}}}{{{""=""}}}{{{4^n-1}}}

{{{16384}}}{{{""=""}}}{{{4^n}}}

{{{4^7}}}{{{""=""}}}{{{4^n}}}

{{{7}}}{{{""=""}}}{{{n}}}

Substitute in 

{{{sum(a[1]*r^k,k=1,n)}}}

{{{sum((expr(3/2)*4^k),k=1,n)}}}

Edwin</pre>