Finding nth term of these numbers...please help...I think I have the wrong function?

Question

Still working on this darn paper....

I was given these numbers and ask to find the nth term.

6 8 12 18 26 36 48 62

Somehow I came up with this function:

x^2 - x + 6

She marked it as correct but now that I am trying to put the entire paper together I just can't seem to get it to workout.
Is this correct or have I totally screwed up?

Thanks!

Answer 1

It looks correct to me:

f(x) = x² - x + 6
f(1) = 1² - 1 + 6 = 6
f(2) = 2² - 2 + 6 = 8
f(3) = 3² - 3 + 6 = 12
...
f(7) = 7² - 7 + 6 = 48
f(8) = 8² - 8 + 6 = 62
f(9) = 9² - 9 + 6 = 78
etc.

Answer 2

I does work Because n=1 is 6, n=2 is 8 and if you plug 1 in for x you get 6, 2 in you get 8.

Changing difference formula
This is a bit more complicated, and is applied to sequences where the difference between each number is not a constant, as in the sequence: 3, 5, 9, 15, 24, ... The trick here is to find the difference increase - so, the difference between 3 and 5 is 2, the difference between 5 and 9 is 4, the difference between 9 and 15 is 6... see the pattern emerging? The difference increase is 2. So, we use the formula -
a + (n-1)d + 0.5(n-1)(n-2)C

Where -
d = the first difference (2 in the sequence above).
C = The difference increase.
a = the first term.

So, an example of its use, using the sequence above -

a + (n-1)d + 0.5(n-1)(n-2)C
3 + (n-1)2 + 0.5(n-1)(n-2)2
3 + 2n - 2 + n2 - 2n - n + 2> 3 + n2 - n


Proof, by finding the 4th term:


3 + n2 - n
3 + 16 - 4

Answer 3

The differences are in arithmetic progression 2,4,6
S1=6
S2=6+2
S3=6+(2+4)
Therefore
Sn=6+{(n-1)/2[4+(n-2)2]
=6+(n-1)*n
=6+n^2-n
(Sum of n terms of an arithmetic series n/2{2a+(n-1)d} where a is the first term and d is the difference. Since we are starting with 2 in the 2nd term and the difference is 2 substitute n-1 for n.

Answer 4

Your function is correct.
We know it must be a quadratic function, so Let's make the function an^2+bn+c=f(n)
Plug in the numbers,
a+b+c=6
4a+2b+c=8
9a+3b+c=12
Use matrix to solve and get n^2-n+6=f(n)

Answer 5

UR formula is correct.

Answer 6

x^2-x+6
x^2-3x+2x+6
x(x-3)+2(x-3)
(x-3)(x+2)