SOLUTION: what is the nth term of the sequence 0,6,14,24,36,50,...

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Question 1187173: what is the nth term of the sequence 0,6,14,24,36,50,...
Answer by MathLover1(20850) About Me  (Show Source):
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0,6,14,24,36,50,.......

find differences
0 ... 6 ... 14 ... 24 ... 36 ... 50
... 6 ... 8+... +10 ... 12 ... 14
......2 ......+2 ......2 ......2

second differences are same, nth term formula is quadratic

a%5Bn%5D=a%2An%5E2+%2B+b%2An+%2Bc

a%5B1%5D=0
a%5B2%5D=6
a%5B3%5D+=+14

0=a%2A1%5E2+%2B+b+%2A1+%2Bc
0=a%2B+b++%2Bc..........solve for c
c=-a-b....eq.1

6=a%2A2%5E2+%2B+b+%2A2+%2Bc
6=4a%2B+2b+%2Bc............solve for c
c=+6-4a-2b...........eq.2
14=a%2A3%5E2+%2B+b+%2A3+%2Bc
14=9a%2B+3b+%2Bc............solve for c
c=+14-9a-3b...........eq.3

from eq.1 and eq.2 we have
-a-b=6-4a-2b.......solve for b
2b-b=6-4a%2Ba
b=6-3a ..............eq.1))

from eq.1 and eq.3 we have
-a-b=14-9a-3b.....solve for b
3b-b=14-9a%2Ba
2b=14-8a.........divide by 2
b=7-4a............eq.2))

equal eq.1)) and eq.2))
6-3a=7-4a....solve for+a
4a-3a=7-6
a=1

go to
b=6-3a ..............eq.1), substitute a
b=6-3%2A1+
b=3
go to
c=-a-b....eq.1, substitute a+and b
c=-1-3
c=-4
nth term formula is:
a%5Bn%5D+=n%5E2+%2B+3+n+-+4
or
a%5Bn%5D+=+%28n+-+1%29+%28n+%2B+4%29+