Question

Question:Fibonacci

Answer 1

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

Answer 2

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.

__

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