look at the differences between successive terms:
-1 -3 -5 -7
so, the next will be 15-9=6
the wonders of Wolfram
did you notice that consecutive differences are
-1, -3, -5, -7 ?
this forms an arithmetic sequence, so a quadratic will act as the defining function.
let the sequence be defined by
f(n) = an^2 + bn + c
for (1,31) ---> a + b + c = 31 , #1
for (2,30) ---> 4a + 2b + c = 30 , #2
subtract these two equations
3a + b = -1
for (3,27) ---> 9a + 3b + c = 27 , #3
#3 - #2 ---> 5a + b = -3
5a + b = -3
3a + b = -1
2a = -2
a = -1
-3 +b = -1
b = 2
in #1: a+b+c=31
-1+2+c = 31
c = 30
the defining function is
fn) = -n^2 + 2n + 30
(just like Wolfram said)
I guess I didn't read the question too carefully, and failed to answer it fully.
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