First differences start at 1 and increase by 2.
Whenever analyzing any sort of sequence, it is usually helpful to look at the differences between terms. Here, they are ...
... 1, 3, 5, 7
This set of numbers clearly increases by 2 from one to the next. That is, second differences are 2.
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When 2nd differences are constant, the pattern can be described by a 2nd-degree polynomial. In this case, that is
... n² -2n +2 . . . . for n = 1, 2, 3, ...
Answer:
Quadratic sequence with
nth term = n^2 - 2n + 2.
Step-by-step explanation:
1,2,5,10,17
The differences between terms are
1,3,5,7
and the second differences are
2,2,2
This is a quadratic sequence with first term n^2 ( where n = sequence number).
List the original sequence and the values of n^2:-
1 2 5 10 17
1 4 9 16 25
If we subtract the first list from the second we get
0 2 4 6 8 which is arithmetic with common difference 2 and first term 0
The nth term of this is 2(n - 1)
So the explicit formula for the nth term of the original sequence is
n^2 - 2(n - 1)
= n^2 - 2n + 2
