SOLUTION: What is the recursive formula of 4, 4¼, 4½, 5

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Question 1164133: What is the recursive formula of 4, 4¼, 4½, 5
Found 2 solutions by Edwin McCravy, ikleyn:
Answer by Edwin McCravy(20064) About Me  (Show Source):
You can put this solution on YOUR website!

4, 4¼, 4½, 5
4, 4.25, 4.5, 5

a%5Bn%5D=p%2Aa%5Bn-1%5D%2Bq%2Aa%5Bn-2%5D,  a(1)=4, a(2)=4.25, n=3,4,5,...

 a%5B3%5D=p%2Aa%5B2%5D%2Bq%2Aa%5B1%5D

4.5=p%2A4.25%2Bq%2A4

a%5B4%5D=p%2Aa%5B3%5D%2Bq%2Aa%5B2%5D

5=p%2A4.5%2Bq%2A4.25 

Solve this system:

system%284.5=p%2A4.25%2Bq%2A4%2C5=p%2A4.5%2Bq%2A4.25%29 

Multiply both equations through by 100 to clear decimals

system%28450=p%2A425%2Bq%2A400%2C500=p%2A450%2Bq%2A425%29

Divide both equations through by 25

system%2818=p%2A17%2Bq%2A16%2C20=p%2A18%2Bq%2A17%29

system%2817p%2B16q=18%2C18p%2B17q=20%29

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

system%28-306p-288q=-324%2C306p%2B289q=340%29

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

an = -14an-1 + 16an-2, a(1)=4, a(2)=4.25, n=3,4,5,...

Edwin

Answer by ikleyn(52903) About Me  (Show Source):