Question 1094569
If  25/k = 1^2 -2^2/5 + 3^2/5^2 - 4^2/5^3+.......infinity , 
then what is the value of k?
<pre>
{{{25/k}}}{{{""=""}}}{{{1^2 -2^2/5^1 + 3^2/5^2 - 4^2/5^3+5^2/5^4-""*""*""*""}}}

{{{25/k}}}{{{""=""}}}{{{1 -2^2/5^1 + 3^2/5^2 - 4^2/5^3+5^2/5^4-""*""*""*""}}}

Multiply through by 5

{{{125/k}}}{{{""=""}}}{{{5 -2^2 + 3^2/5^1 - 4^2/5^2+5^2/5^3-6^2/5^4+""*""*""*""}}}

Add the two preceding sequences:

{{{150/k}}}{{{""=""}}}{{{6-4 + (3^2-2^2)/5^1 - (4^2-3^2)/5^2+(5^2-4^2)/5^3-(6^2-5^2)/5^4+""*""*""*""}}}

{{{150/k}}}{{{""=""}}}{{{2 + (9-4)/5^1 - (16-9)/5^2+(25-16)/5^3-(36-25)/5^4+""*""*""*""}}}

{{{150/k}}}{{{""=""}}}{{{2 + 5/5^1 - 7/5^2+9/5^3-11/5^4+""*""*""*""}}}

{{{150/k}}}{{{""=""}}}{{{2 + 1 - 7/5^2+9/5^3-11/5^4+""*""*""*""}}}

{{{150/k}}}{{{""=""}}}{{{3 - 7/5^2+9/5^3-11/5^4+""*""*""*""}}}

Multiply by 5

{{{750/k}}}{{{""=""}}}{{{15 - 7/5^1+9/5^2-11/5^3+13/5^4-""*""*""*""}}}

Add the two preceding sequences:

{{{900/k}}}{{{""=""}}}{{{18 - 7/5+2/5^2-2/5^3+2/5^4-""*""*""*""}}}

{{{900/k}}}{{{""=""}}}{{{83/5+2/5^2-2/5^3+2/5^4-""*""*""*""}}}

From the 2nd term on the right onward is an infinite geometric series
 with  {{{a[1]=2/5^3}}} and {{{r=-1/5}}}, so we use the formula:

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

{{{900/k}}}{{{""=""}}}{{{83/5+(2/5^2)/(1-(-1/5))}}}

{{{900/k}}}{{{""=""}}}{{{83/5+(2/25)/(1+1/5)}}}

{{{900/k}}}{{{""=""}}}{{{83/5+(2/25)/(6/5)}}}

{{{900/k}}}{{{""=""}}}{{{83/5+(2/25)*(5/6)}}}

{{{900/k}}}{{{""=""}}}{{{83/5+1/15}}}

{{{900/k}}}{{{""=""}}}{{{50/3}}}

Divide both sides by 50

{{{18/k}}}{{{""=""}}}{{{1/3}}}

Cross-multiply:

{{{k}}}{{{""=""}}}{{{54}}}

Edwin</pre>