Question on this math method?

Question

First, your sequence is: 1, 1, 5, 13, 25, 41, 61

Take the differences between adjacent terms:0, 4, 8, 12, 16, 20

Take the difference again: 4, 4, ,4 ,4, 4

Notice that all the second differences are the same. This means the polynomial will be second order so the sequence must look like:

a*n^2 + b*n + c

Now its a matter of solving for a, b, and c. If you call this last difference (here its 4) k and your highest exponent m (here that is 2) then a = k/m!

To get the next coefficient (b) subtract a*n^2 from each term so the terms become:

1 - 2*1^2 = -1
1 - 2*2^2 = -7

You don't need to do the whole sequence, you only need to do one term more than the exponent you are trying to solve for. you get -6, -6, -6, -6, etc.
Using the formula for coefficients above b = -6/1! = -6

So the sequence formula so far is 2*n^2 - 6*n + c

To find c, just evaluate for n=1: 2*1^2 - 6*1 + c = 1
So: c=5
---------------------------------
why is difference -6 not 6? why is formula for c equal to 1?

Answer 1

Why is the difference -6 not 6? Well, 0, 4, 8 has difference 4 since 8 - 4 = 4 - 0 = 4. Then -1, -7, -13 has difference -6 since -13 - (-7) = -7 - (-1) = -6.

Why is formula for c equal to 1? Because the first number of the sequence is 1.

Answer 2

at n=1, we get a value of 1, so substitute 1 for n and set the equation = to 1:

2*n^2 - 6*n + c = 1
2*1^2 - 6*1 + c = 1
2 - 6 + c = 1
-4 + c = 1
c = 5

So your final formula would be f(n) = 2n^2 - 6n + 5

n = 1: 2(1^2) - 6(1) + 5 = 2 - 6 + 5 = 1
n = 2: 2(2^2) - 6(2) + 5 = 8 - 12 + 5 = 1
n = 3: 2(3^2) - 6(3) + 5 = 18 - 18 + 5 = 5
n = 4: 2(4^2) - 6(4) + 5 = 32 - 24 + 5 = 13
n = 5: 2(5^2) - 6(5) + 5 = 50 - 30 + 5 = 25

as you can see, the formula works!

Hope this helps! :)

Answer 3

nth term = an^2 + bn + c
then a + b + c = 1 .............................(1)
4a + 2b + c = 1 ...........................(2)
9a + 3b + c = 5 ...........................(3)
(1) and (2) give 3a + b = 0 Or b = -- 3a .........(4)
(2) and (3) give 5a + b = 4 Or b = 4 -- 5a ......(5)
(4) and (5) give -- 3a = 4 -- 5a Or 2a = 4 => a = 2
then b = 4 -- 5a = 4 -- 5(2) = -- 6
c = 1 -- a -- b = 1 -- 2 + 6 = 5
then nth term = an^2 + bn + c = 2n^2 -- 6n + 5

Answer 4

I think the step to find b is suspicious, since you are using two terms that are the same.