Question

Question #69d74

Answer 1

`a_n=n^2+3n+1`

Explanation:

A quadratic sequence is in the form of: `an^2+bn+c`.

First, we need to confirm that this sequence is quadratic, and this is done by finding the second difference.

`5,11,19,29,...`

First difference: `6,8,10,...` `(+2)` each time

Second difference: `2,2,2`

If we divide the second difference by `2`, we get the value of `a`.

`2/2=1`, so `a=1`

So far we got `n^2`.

Right now, we need to plug in values for `n^2` and compare them with the above sequence.

`n^2=1,4,9,16,...`

`S_n=5,11,19,29,...`

We see that `1+4=5`, `4+7=11`, `9+10=19`, and so on.

The differences are `4,7,10,...`, and it increases by `3` each time. So, that is the value of `b`.

So far, we have: `n^2+3n`.

Final step, we plug in values for `n^2+3n` and compare them with the original sequence.

`S_n=5,11,19,29,...`

`n^2+3n=4,10,18,28,...`

We see that `4+1=5`, `10+1=11`, `18+1=19`, and so on.

So, the common difference is `1`, and that is the value of `c`.

Final step is to put `a=1, b=3, c=1` into the form of `an^2+bn+c`.

So, the `n_"th"` term of this sequence will be given by: `n^2+3n+1`.

For more practice, visit: https://owlcation.com/stem/Quadratic-Sequences-The-nth-term-of-a-quadratic-number-sequence

I hope this helps!

Answer 2

`a_n=A((n-1),(0))+B((n-1),(1))+C((n-1),(2))`

Explanation:

Hmm...

Let's find the difference between each term.

From `{5,11,19,29...}`

The difference between the terms go like this:

`{6,8,10...}`

Let's find the difference between each difference:

`{2,2,2...}`

I have put it like this:

Now, there is a formula for a sequence where the second difference is constant, and the first difference forms an arithmetic sequence.

The formula is this:

`a_n=A((n-1),(0))+B((n-1),(1))+C((n-1),(2))`

This is called Newton's Little Formula for the nth term.

Note: `((n-1),(k))=(n-1)Ck` (Another way to write combination)

For example, if you wanted to know the hundredth term in this sequence, then it goes like this:

`a_100=5((99),(0))+6((99),(1))+2((99),(2))`

`a_100=5*1+6*99+2*4851`

`a_100=5+594+9702`

`a_100=10301`

It would've been much harder if you were to do this by writing out the terms!