Question

What is the nth term of the sequence -2 -8 -18 -32 -50

Answer 1

-2n²

Explanation:

Using the image attached:
We see that the second differences (blue ones) are all equal so we conclude that this is a quadratic sequence.
The quadratic sequence has the form:

`T_(n) = an^(2) +bn +c`

To find the value of a we just divide second difference ( -4 ) by 2:

• 
  `a = (-4)/(2)= -2`

Now we have:

• 
  `T_(n)= -2n^(2) +bn+c`

Substitute
`n_(1)` and
`n_(2)` into the equation above:

• 
  `T_(1) = -2(1)^(2) +b(1) +c`

• 
  `T_(2) = -2(2)^(2) +b(2)+c`

Since
`T_(1)` = -2 and
`T_(2)` = -8 , after simplification we have:

• 
  `b +c =0`
• 
  `2b+c=0`

The solution of this system is:

• b=0, c =0

The general term is :

`T_(n) = -2n^(2)`