determine the rule for the nth term

Question

The first 5 terms of a sequence are given. 2,6,12,20,30,... Determine the rule for the nth term.

Answer 1

differences between the terms:

2, 6, 12, 20, 30

  4  6   8    10

     2  2   2

 

Since it takes two levels of differences to get to a constant difference (2 in this case), it will be a 2nd order polynomial.

 

Our function will be of the form: A x^2 + B x +C

 

The equations for the first three points is:

A * (1^2) + B * 1 + C = 2

A * (2^2) + B * 2 + C = 6

A * (3^2) + B * 3 + C = 8

 

simplified

  A +  B + C = 2

4A + 2B + C = 6

9A + 3B + C = 12

 

Solving this system of simultaneous equations (3 equations, 3 unknowns - left to the reader) yields A = 1, B = 1 and C = 0.

 

Therefore, the rule is

f(n) = x^2 + x

Answer 2

Hi Nia!  To solve this, it helps to look at the sequence you've been given, alongside each n:

 

n     x

1     2

2     6

3    12

4    20

5    30

 

You can see that each x value is a function of its n and the n that's immediately following.  For example, 6 is equal to 2 (the n for 6) times 3 (the n for 6, plus 1).  What formula can you write that will generalize that example, solving for any x using n and the next n?

Answer 3

Since the first diffrence gives you an arithmetic sequence, you can get the answer directly by adding the sum of an arithmetic series to the first term.

 

2, 6, 12, 20, 30

  4   6   8    10

t(n) = 2 + 4 +...+ 2n = 2(1+2+...+n) = 2(1/2)(1+n)n = n^2 + n

Check:

t(1) = 1^1+1 = 2

t(2) = 2^2+2 = 6

t(3) = 3^2+3 = 12

t(4) = 4^2+4 = 20

t(5) = 5^2+5 = 30