SOLUTION: i need to know the nth term for and how. thanks. 8,43,138,323,628,1083, 1718....

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Question 939169: i need to know the nth term for and how. thanks.
8,43,138,323,628,1083, 1718....
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20855) About Me  (Show Source):
You can put this solution on YOUR website!

first find differences between each two given terms and go until you get all differences equal
8-----43-----138-------323-----628-----1083-----1718....
---35-----95------185------305-----455-----635------
-------60-----90------120-----150-----180------
-----------30-----30------30-------30------=> third differences same, means we have third degree function
first term is 8 and it can be written as 5%2B3
if we have third degree function n%5E3 to get first term if n starts with 1 we can write it as 5n%5E3%2B3
so, general equation for nth term is:

a%5Bn%5D+=+5n%5E3%2B3+ where n=1,2,3,........

check:
a%5Bn%5D+=+5n%5E3%2B3+ if n=1
a%5B1%5D+=+5%281%29%5E3%2B3+
a%5B1%5D+=+5%281%29%2B3+
a%5B1%5D+=+5%2B3+
a%5B1%5D+=+8+

a%5Bn%5D+=+5n%5E3%2B3+ if n=2
a%5B2%5D+=+5%282%29%5E3%2B3+
a%5B2%5D+=+5%288%29%2B3+
a%5B2%5D+=+40%2B3+
a%5B2%5D+=+43+

a%5Bn%5D+=+5n%5E3%2B3+ if n=3
a%5B3%5D+=+5%283%29%5E3%2B3+
a%5B3%5D+=+5%2827%29%2B3+
a%5B3%5D+=+135%2B3+
a%5B3%5D+=+138+




Answer by Edwin McCravy(20081) About Me  (Show Source):
You can put this solution on YOUR website!
Here's the way I did it.  Not with a difference table.

Let the nth term be an.

I noticed first that the last digits go 8,3,8,3,8,3,8.

Since the last digits differ by ±5, I noticed that if 
we subtract 3 from each one, the last digits of the new 
sequence will go 5,0,5,0,5,0 and therefore all be 
multiples of 5.

Let's subtract 3 from each, and we get a sequence
which has nth term a%5Bn%5D-3 

5, 40, 135, 320, 625, 1080, 1715

Let's divide them by 5 and get a sequence with 
nth term %28a%5Bn%5D-3%29%2F5

1, 8, 27, 64, 125, 216, 343

Aha. that is the sequence of cubes which has
nth term n%5E3.

So we have the equation 

%28a%5Bn%5D-3%29%2F5%22%22=%22%22n%5E3

a%5Bn%5D-3%22%22=%22%225n%5E3

a%5Bn%5D%22%22=%22%225n%5E3%2B3

Edwin