SOLUTION: What is the formula for this sequence: 21, 45, 77, 117, 165 ?

Question 847870: What is the formula for this sequence:
21, 45, 77, 117, 165 ?

Answer by swincher4391(1107) (Show Source): You can put this solution on YOUR website!
45 -21 = 24
77-45 = 32
117 - 77 = 40...
etc. It seems as though our difference is increasing by 8 each time.
By knowing that the 2nd difference is 8, we know that the function is of the form 4n^2 + something. In general, an^2 has second difference 2a.
We also realize that the difference between a(0) and a(1) is 16.
So a(0)= 5.
So we have 4n^2 + b*n + 5
This should be enough to figure out what b is.
Take a_2 for instance.
4(2)^2 + b*2 + 5 = 45
16 + 2b + 5 = 45
2b = 24
b = 12
So our entire function a(n) = 4n^2 + 12n + 5 <---- ANSWER
We check our answer:
a(1) = 4 + 12 + 5 = 21
a(2) = 4*4 + 24 + 5 = 45
a(3) = 4*9 + 36 + 5 = 77
a(4) = 4*16 + 48 + 5 = 117
a(5) = 4*25 + 60 + 5 = 165
Just for fun a(10) = 4*100 + 12*10 + 5 = 525. Simple.
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