Question 1010974: Find both an explicit formula and recursive formula for the nth term for arithmetic sequence 76,70,63
Answer by mathmate(429)   (Show Source):
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Question:
Find both an explicit formula and recursive formula for the nth term for arithmetic sequence 76,70,63
Solution:
The given sequence is NOT an arithmetic since the difference increases from 6 to seven.
Assuming the sequence is a quadratic function.
A. Recursive formula
Recall that the difference increases by one for successive terms, the recursive formula is of the form:
T(n)=T(n-1)-(n+k),
Since T(1)=76, T(2)=T(1)-(n+k)=> 70=76-(2+k) => k=4
therefore
T(n)=T(n-1)-(n+4), with T(1)=76
Check: T(2)=76-(2+4)=70; T(3)=70-(3+4)=63, ok.
B. Explicit formula:
We have only three known terms, and knowing that the sequence is not arithmetic, we will assume the sequence is quadratic, of the form:
from which we can substitute for n=1, 2 and 3 to get
76=a+b+c
70=4a+2b+c
63=9a+3b+c
from which we can readily solve (by elimination) to get
 ,  , 
Hence the explicit formula for T(n) is
Check:
T(1)=-0.5-4.5+81=76
T(2)=-2-9+81=70
T(3)=-4.5-13.5+81=63
All satisfied.
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