SOLUTION: Please help with the following. It would be greatly appreciated! Given quadratic pattern: 4;9;X;37 Determine X and Tn

Algebra ->  Finite-and-infinite-sets -> SOLUTION: Please help with the following. It would be greatly appreciated! Given quadratic pattern: 4;9;X;37 Determine X and Tn      Log On


   

Question 926793: Please help with the following. It would be greatly appreciated!
Given quadratic pattern:
4;9;X;37
Determine X and Tn
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
Given quadratic pattern:
4;9;x;37
quadratic:an%5E2+%2B+bn+%2B+c+=+y if y=T%5Bn%5D where n starts with 1, we have
plug in T%5B1%5D=4, means first term, n=1

a%281%29%5E2+%2B+b+%2B+c+=+4
a+%2B+b+%2B+c+=+4............. [1]
T%5B2%5D=9
a%282%29%5E2+%2B+2b+%2B+c+=+9
4a+%2B+2b+%2B+c+=+9.......... [2]
T%5B4%5D=37
a%284%29%5E2+%2B+4b+%2B+c+=+37
16a+%2B+4b+%2B+c+=+37............ [3]
if you subtract [1] from [2], you get
4a+%2B+2b+%2B+c-%28a+%2B+b+%2B+c%29+=+9-4
4a+%2B+2b+%2B+c-a+-b+-c+=+5
3a+%2B+b++=+5.....solve for b
b++=+5-3a
and [3] - [2]
16a+%2B+4b+%2B+c-%284a+%2B+2b+%2B+c%29+=+37-9
16a+%2B+4b+%2B+c-4a+-2b+-+c+=+28
12a+%2B2b+=+28................plug in b++=+5-3a
12a+%2B2%285-3a%29+=+28...solve for a
12a+%2B10-6a+=+28
6a+=+28-10
6a+=18
a+=+3, ...now find b
b++=+5-3%2A3
b++=+5-9
so b+=+-4 and .now find c
a+%2B+b+%2B+c+=+4............. [1]
3+-4+%2B+c+=+4
-1+%2B+c+=+4
c+=+4%2B1
c+=+5
then third term will be:
3%283%5E2%29+-+4%283%29+%2B+2+=+x
x+=+20+
quadratic pattern:
4;9;20;37
T%5Bn%5D=3n%5E2+-4n+%2B+5+ where n=1,2,3,....