SOLUTION: 5,5,6,8,11,15 what is the 20th term and 21th

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Question 512608: 5,5,6,8,11,15 what is the 20th term and 21th
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
          1st          2nd  
n  an  differences differences
1   5     0           1               
2   5     1           1
3   6     2           1  
4   8     3           1  
5  11     4
6  15     

Since the 2nd differences are all constant we try a 2nd
degree polynomial in n

a%5Bn%5D+=+An%5E2%2BBn%2BC



Simplifying:

system%285=+A%2BB%2BC%2C+%0D%0A++++++++++5=+4A%2B2B%2BC%2C+%0D%0A++++++++++6=+9A%2B3B%2BC%29system%28A%2BB%2BC=5%2C+%0D%0A++++++++++4A%2B2B%2BC=5%2C+%0D%0A++++++++++9A%2B3B%2BC=6%29

Solve that system and get A = 1%2F2, B = -3%2F2, C = 6

So the equation

a%5Bn%5D+=+An%5E2%2BBn%2BC

becomes:

a%5Bn%5D+=+expr%281%2F2%29n%5E2-expr%283%2F2%29n%2B6

a%5Bn%5D+=+expr%281%2F2%29n%5E2-expr%283%2F2%29n%2B12%2F2

a%5Bn%5D+=+%28n%5E2-3n%2B12%29%2F2

Substituting n = 20

a%5B20%5D+=+%2820%5E2-3%2A20%2B12%29%2F2

a%5B20%5D+=+%28400-60%2B12%29%2F2

a%5B20%5D+=+352%2F2

a%5B20%5D+=+176

Substituting n = 21

a%5B21%5D+=+%2821%5E2-3%2A21%2B12%29%2F2

a%5B21%5D+=+%28441-63%2B12%29%2F2

a%5B21%5D+=+390%2F2

a%5B21%5D+=+195


Answers: a%5B20%5D+=+176, a%5B21%5D+=+195

Edwin