Tutor @ikleyn is correct, in that the problem AS POSTED cannot be solved mathematically; ANY number inserted in the given sequence forms a valid sequence.
However, there is some useful mathematics to be practiced in solving the problem if we assume the given sequence is formed by a polynomial function.
We are given 5 points of a polynomial function: f(1)=444; f(2)=456; f(3)=471; f(5)=498; and f(6)=519. Our objective is to find the value of f(4).
n points can be fitted with a unique polynomial of degree (n-1). Since we have 5 points, we can fit them with a polynomial of degree 4.
The general polynomial of degree 4 is f(x) = ax^4+bx^3+cx^2+dx+e. I used the matrix capability of by TI-84 calculator to find that polynomial and evaluate it for x=4 to find the answer.
However, when the problem only asks to find the missing number, we don't need to find the function itself. We can use the method of finite differences.
Here is a preliminary look at the method of finite differences using a simple example.
Consider the quadratic polynomial function f(x)=3x^2-5x+2. That function has the values
f(1) = 0
f(2) = 4
f(3) = 14
f(4) = 30
We will use these four function values to show that the sequence is generated by a polynomial function of degree 2. (We could then continue with the process to determine the actual function; however that part of the process is not needed for the demonstration below of how to find the missing number in your problem.)
Here is what the method of finite differences looks like:
0 4 14 30 < the given sequence
4 10 16 < the "first differences"
6 6 < the "second differences"
The constant second differences tell us that the sequence can be produced by a polynomial of degree 2.
(If you know basic differential calculus, this is because the second derivative of a quadratic polynomial is a constant. In fact, if you know that calculus, you know that the second derivative of a polynomial with leading term is ax^2 will be the constant 2a. In my simple example, the leading coefficient of the polynomial is 3, and the constant second difference is 2*3=6.)
In general, if a sequence is produced by a polynomial of degree n, then the row of n-th differences will be constant.
We are now ready to find the missing number in your problem, assuming the sequence is defined by a polynomial of degree 4, which means the 4th differences must be constant.
444 456 471 x 498 519 < the given sequence
12 15 x-471 498-x 21 < 1st differences
3 x-486 969-2x x-477 <2nd differences
x-489 1455-3x 3x-1446 <3rd differences
1944-4x 6x-2901 <4th differences
The row of 4th difference must be constant, so...
ANSWER: The missing number is 484.5
This result was confirmed with the matrix work I did on my TI-84 calculator.