Question 1154626
The third term of a geometric sequence is {{{12}}}

{{{a[3]=12}}}

 and the fifth term is {{{163}}}

{{{a[5]=163}}}

the formula for an  nth term of this sequence:

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


from given above we have

{{{12=a[1]*r^(3-1)}}}

{{{12=a[1]*r^2}}}.......eq.1

and 

{{{163=a[1]*r^(5-1)}}}

{{{163=a[1]*r^4}}}.......eq.2


solve the system:

{{{12=a[1]*r^2}}}.......eq.1
{{{163=a[1]*r^4}}}.......eq.2
------------------------------------

{{{12=a[1]*r^2}}}.......eq.1.....solve for {{{a[1]}}}
{{{12/r^2=a[1]}}}.......eq.1a..........substitute in eq.2


{{{163=(12/r^2)*r^4}}}.......simplify

{{{163=12r^2}}}

{{{163/12=r^2}}}

{{{r=sqrt(163/12)}}}-> exact solution


go to eq.1a

{{{12/r^2=a[1]}}}............substitute {{{r}}}

{{{a[1]=12/(sqrt(163/12))^2}}}

{{{a[1]=12/(163/12)}}}

{{{a[1]=144/163}}}-> first term


then, the formula for an  nth term of this sequence:


{{{a[n]=(144/163)*(sqrt(163/12))^(n-1)}}}


 the value of {{{a[25]}}} will be


{{{a[25]=(144/163)*(sqrt(163/12))^(25-1)}}}


{{{a[25]=(144/163)*(sqrt(163/12))^24}}}


{{{a[25]=2158060662623960090407387/61917364224}}}


{{{a[25] =34853884522872& 58007876059/61917364224}}}-> exact solution


{{{a[25] }}} ≈{{{34853884522873}}}