Question 847871: What is the formula for this sequence:
21, 45, 77, 117, 165 ?
Found 2 solutions by hamsanash1981@gmail.com, swincher4391: Answer by hamsanash1981@gmail.com(151) (Show Source):
You can put this solution on YOUR website! 21, 21+8*3,45+8*4, 77+8*5, 117+8*6, 165+8*7
21,45,77, 117,165, 221.
Answer by swincher4391(1107) (Show Source):
You can put this solution on YOUR website! Let's look at some things. What is the difference between these numbers?
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
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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