9514 1404 393
Answer:
n² -n +1
Step-by-step explanation:
First differences of the terms of the sequence are ...
3-1 = 2
7-3 = 4
13-7 = 6
21-13 = 8
Second differences are ...
4-2 = 2
6-4 = 2
8-6 = 2
In a quadratic sequence the coefficient of the squared term is half of the second difference value. Here, that means the squared term will be (2/2)n² = n².
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We can find the other terms of the quadratic expression by considering the differences between n² and the actual sequence.
The sequence of n² terms is ...
1, 4, 9, 16, 25
When we subtract these from the actual sequence, we get ...
1-1 = 0
3-4 = -1
7-9 = -2
13-16 = -3
21-25 = -4
That is, the amount subtracted from n² is one less than the term number.
The expression for the n-th term is ...
an = n² -n +1
