Question

Here are the first five terms of a sequence. 4, 11, 22, 37, 56 Find an expression, in terms of , for the th term of this sequence.

Answer 1

`a_n = 2n^2 + n + 1`

Explanation:

4, 11, 22, 37, 56

11 - 4 = 7

22 - 11 = 11

37 - 22 = 15

56 - 37 = 19

After the first difference, 11 - 4 = 7, each difference is 4 more than the previous difference.

Difference of differences:

11 - 7 = 4

15 - 11 = 4

19 - 15 = 4

Since we need the difference of differences to find a constant, this must be a second degree function.

`a_1 = 4 = 2^2 + 1(0)`

`a_2 = 11 = 3^2 + 2 = 3^2 + 2(1)`

`a_3 = 22 = 4^2 + 6 = 4^2 + 3(2)`

`a_4 = 37 = 5^2 + 12 = 5^2 + 4(3)`

`a_5 = 56 = 6^2 + 20 = 6^2 + 5(4)`

`a_n = (n + 1)^2 + (n)(n - 1)`

`a_n = (n + 1)^2 + (n)(n - 1)`

`a_n = n^2 + 2n + 1 + n^2 - n`

`a_n = 2n^2 + n + 1`