Determine the general term, tn, for the following sequence. 8,20,36,56

Question

Determine the general term, tn, for the following sequence.      8,20,36,56

Bob Huerta · Accepted Answer

Check the difference in each term
After the second term the difference is the same, so n-th term of this series would be polinomial with degree 2.
So,

t_{n} = a n^{2} + b n + c

t_{1} = 8

,

t_{2} = 20

,

t_{3} = 36

but

n = 1

,

t_{1} = 8

as we have below

a (1) + b (1) + c = 8

a + b + c = 8

(1)

t_{2} = 20

,

n = 2

, so we have

a {(2)}^{2} + b (2) + c = 20

4 a + 2 b + c = 20

(2)

t_{3} = 36

,

n = 3

a {(3)}^{2} + b (3) + c = 36

9 a + 3 b + c = 36

(3)
Solve all the equations. Subtract equation (1) from (2)

4 a + 2 b + c - (a + b + c) = 20 - 8

3 a + b = 12

(4)
Now, subtract equation (2) from (3)

9 a + 3 b + c - (4 a + 2 b + c) = 36 - 20

5 a + b = 16

(5)
So,

5 a + b - (3 a + b) = 16 - 12

2 a = 4

a = 2

Calculate b

5 (2) + b = 16

b = 6

From equation (1) find c

2 + 6 + c = 8

c = 0

Consider n-th term

t_{n} = a n^{2} + b n + c

and

a = 2

,

b = 6

,

c = 0

t_{n} = (2) (n^{2}) + 6 (n) + 0

t_{n} = 2 n^{2} + 6 n

is the answer

Determine the general term, t_n, for the following sequence.

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