Question 1164133
<pre>

4, 4¼, 4½, 5
4, 4.25, 4.5, 5

{{{a[n]=p*a[n-1]+q*a[n-2]}}},  a(1)=4, a(2)=4.25, n=3,4,5,...

 {{{a[3]=p*a[2]+q*a[1]}}}

{{{4.5=p*4.25+q*4}}}

{{{a[4]=p*a[3]+q*a[2]}}}

{{{5=p*4.5+q*4.25}}} 

Solve this system:

{{{system(4.5=p*4.25+q*4,5=p*4.5+q*4.25)}}} 

Multiply both equations through by 100 to clear decimals

{{{system(450=p*425+q*400,500=p*450+q*425)}}}

Divide both equations through by 25

{{{system(18=p*17+q*16,20=p*18+q*17)}}}

{{{system(17p+16q=18,18p+17q=20)}}}

To eliminate p multiply first equation through by -18
and second equation through by 17

{{{system(-306p-288q=-324,306p+289q=340)}}}

Add term by term

q = 16

Substitute 16 for q in

   17p+16q = 18
17p+16(16) = 18
   17p+256 = 18
       17p = -238
         p = -14

So the recursion formula is

a<sub>n</sub> = -14a<sub>n-1</sub> + 16a<sub>n-2</sub>, a(1)=4, a(2)=4.25, n=3,4,5,...

Edwin</pre>