SOLUTION: NEED HELP How do I make a formula for an increasingly decreasing sequence? eg: 50,49,47,44,40,35,... and how do I find the next three terms of the sequence?

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Question 1164687: NEED HELP
How do I make a formula for an increasingly decreasing sequence?
eg: 50,49,47,44,40,35,...
and how do I find the next three terms of the sequence?
Found 2 solutions by Theo, Edwin McCravy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
i found what seems to work.

formula would be An = A1 - sum of (i - 1) for i = 1 to n.

formula would work like this:

when n = 1, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A1 = 50 minus sum of (i - 1) for i = 1 to 1 which becomes:
A1 = 50 minus 0 which becomes:
A1 = 50

when n = 2, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A2 = 50 minus sum of (i - 1) for i = 1 to 2 which becomes:
A2 = 50 minus 0 minus 1 which becomes:
A2 = 50 minus 1 which becomes:
A2 = 49

when n = 3, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A3 = 50 minus sum of (i - 1) for i = 1 to 3 which becomes:
A3 = 50 minus 0 minus 1 minus 2 which becomes:
A3 = 50 minus 3 which becomes:
A3 = 47

when n = 4, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A4 = 50 minus sum of (i - 1) for i = 1 to 4 which becomes:
A4 = 50 minus 0 minus 1 minus 2 minus 3 which becomes:
A4 = 50 minus 6 which becomes:
A4 = 44

when n = 5, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A5 = 50 minus sum of (i - 1) for i = 1 to 5 which becomes:
A5 = 50 minus sum of 0 minus 1 minus 2 minus 3 minus 4 which becomes:
A5 = 50 minus 10 which becomes:
A5 = 40

when n = 6, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A6 = 50 minus sum of (i = 1) for i = 1 to 6 which becomes:
A6 = 50 minus 0 minus 1 minus 2 minus 3 minus 4 minus 5 which becomes:
A6 = 50 minus 15 which becomes:
A6 = 50 - 15 which becomes:
A6 = 35

you can see that this formula gets you the number you are looking for, which are:
A1 = 50
A2 = 49
A3 = 47
A4 = 44
A5 = 40
A6 = 35

finding the next 3 terms in the sequence is simply applying the formula.

when n = 7, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A7 = 50 minus sum of (i - 1) for i = 1 to 7 which becomes:
A7 = 50 minus 0 minus 1 minus 2 minus 3 minus 4 minus 5 minus 6 which becomes:
A7 = 50 minus 21 which becomes:
A7 = 29

when n = 8, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A8 = 50 minus sum of (i - 1) for i = 1 to 8 which becomes:
A8 = 50 minus 0 minus 1 minus 2 minus 3 minus 4 minus 5 minus 6 minus 7 which becomes:
A8 = 50 minus 28 which becomes:
A8 = 22

when n = 9, An = A1 - sum of (i - 1) for i = 1 to n becomes:
A9 = 50 minus sum of (i - 1) for i = 1 to 9 which becomes:
A9 = 50 minus 0 minus 1 minus 2 minus 3 minus 4 minus 5 minus 6 minus 7 minus 8 which becomes:
A9 = 50 minus 36 which becomes:
A9 = 14

to confirm the formula gave you the correct answer, you could do the following.

you are always subtracting 1 more from the previous number than you subtracted from the number preceding that.

50 minus 0 = 50 for n = 1
50 minus 1 = 49 for n = 2
49 minus 2 = 47 for n = 3
47 minus 3 = 44 for n = 4
44 minus 4 = 40 for n = 5
40 minus 5 = 35 for n = 6
35 minus 6 = 29 for n = 7
29 minus 7 = 22 for n = 8
22 minus 8 = 14 for n = 9

the formula here appears to be A.n+1 = A.n - (n-1)
you would start this sequence A1.
from there the formula would get you.
A2 = 50 - 1 = 49
A3 = 49 - 2 = 47
A4 = 47 - 3 = 44
A5 = 44 - 4 = 40
A6 = 40 - 5 = 35
A7 = 35 - 6 = 29
A8 = 29 - 7 = 22
A9 = 22 - 8 = 14

either way, you can see that A7 will be 29 andA8 will be 22 and A9 will be 14.





Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
50,49,47,44,40,35,...
Make the set of first and second differences

50  49  47  44  40  35
  -1  -2  -3  -4  -5
    -1  -1  -1  -1  
  
and since the second differences are constant, we
assume a 2nd degree polynomial in n for the general term:

[If the kth differences are constant, the the general term
will be a polynomial in n of degree k.



Then we substitute the first 3 terms for a1, a2, a3 and 1,2,3 for n

 

or



Solve that system and get 

A = -1/2, B = 1/2, and C = 50

So the general term is



You can find the next three terms either by extending the
difference table:

50  49  47  44  40  35  29  22  14
  -1  -2  -3  -4  -5  -6  -7  -8  
    -1  -1  -1  -1  -1  -1  -1

or by substituting 7, 8, and 9 in the general formula:









Edwin
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