Find a quadratic model for the sequence with the indicated terms.

3, 3, 5, 9, 15, 23, ...

the differences are 0,2,4,6,8,...

Tn+1 = Tn + 2(n-1)
so,
Tn = n^2-3n+5

To find a quadratic model for the sequence, we need to find a quadratic equation that represents the sequence. We can start by listing the differences between consecutive terms to see if there is a pattern:

3, 3, 5, 9, 15, 23, ...
0, 2, 4, 6, 8, ...

The differences between consecutive terms are increasing linearly by 2 each time. This suggests that the original sequence may be represented by a quadratic equation.

To find the quadratic model, we can use the formula y = ax^2 + bx + c, where x represents the position of the term in the sequence (starting from 0) and y represents the term itself.

Using the given terms, we can set up a system of equations:

When x = 0, y = 3: 3 = a(0)^2 + b(0) + c -> 3 = c

When x = 1, y = 3: 3 = a(1)^2 + b(1) + c -> 3 = a + b + c -> 3 = a + b + 3

When x = 2, y = 5: 5 = a(2)^2 + b(2) + c -> 5 = 4a + 2b + 3

Now we can solve this system of equations to find the values of a and b:

From the equation 3 = a + b + 3, we can rearrange and subtract 3 from both sides to get a + b = 0.

From the equation 5 = 4a + 2b + 3, we subtract 3 from both sides to get 2a + b = 2.

Now let's solve the system:

a + b = 0
2a + b = 2

Multiply the first equation by -2 and add it to the second equation to eliminate b:

-2a - 2b = 0
2a + b = 2
--------------
0 = 2

It appears we have a contradiction, which means there might have been an error in our calculations or some information is missing. Please double-check the given terms or let me know if there's any missing information.

To find a quadratic model for the given sequence, we need to find a formula that generates the terms of the sequence.

Looking at the given terms, we can identify the pattern that is followed:
3, 3, 5, 9, 15, 23

If we observe the differences between consecutive terms, we get:
(3 - 3), (5 - 3), (9 - 5), (15 - 9), (23 - 15)
0 , 2 , 4 , 6 , 8

The differences between consecutive terms are increasing by 2 each time. This suggests that the original sequence could be described by a quadratic model.

To confirm this, let's check the differences between these differences:
(2 - 0), (4 - 2), (6 - 4), (8 - 6)
2 , 2 , 2 , 2

The second differences are all equal to 2, indicating that the original sequence is indeed best described by a quadratic model.

Now, let's use this information to construct the quadratic model.
The general form of a quadratic equation is: y = ax^2 + bx + c

Let's assume our quadratic model is: y = ax^2 + bx + c

To find the values of a, b, and c, we can substitute the values of the terms into the quadratic equation and solve the resulting system of equations.

Using the first three terms of the sequence:
3 = a(1^2) + b(1) + c
3 = a(2^2) + b(2) + c
5 = a(3^2) + b(3) + c

Simplifying the above equations, we have:
a + b + c = 3
4a + 2b + c = 3
9a + 3b + c = 5

Solving this system of equations, we find:
a = 1/2, b = 1/2, c = 2

Thus, the quadratic model for the given sequence is:
y = (1/2)x^2 + (1/2)x + 2