document.write( "Question 1208795: 444, 456, 471, ? 498, 519 \n" ); document.write( "

Algebra.Com's Answer #847273 by greenestamps(13200) $\"\"$ $\"About$

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\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "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" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "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.

\n" ); document.write( "Here is a preliminary look at the method of finite differences using a simple example.

\n" ); document.write( "Consider the quadratic polynomial function f(x)=3x^2-5x+2. That function has the values
\n" ); document.write( "f(1) = 0
\n" ); document.write( "f(2) = 4
\n" ); document.write( "f(3) = 14
\n" ); document.write( "f(4) = 30

\n" ); document.write( "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.)

\n" ); document.write( "Here is what the method of finite differences looks like:

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document.write( "   0   4    14    30    < the given sequence\r\n" );
document.write( "     4   10    16       < the \"first differences\"\r\n" );
document.write( "       6     6          < the \"second differences\"

\n" ); document.write( "The constant second differences tell us that the sequence can be produced by a polynomial of degree 2.

\n" ); document.write( "(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.)

\n" ); document.write( "In general, if a sequence is produced by a polynomial of degree n, then the row of n-th differences will be constant.

\n" ); document.write( "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.

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document.write( "  444       456       471         x              498      519  < the given sequence
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document.write( "      12        15        x-471          498-x         21      < 1st differences
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document.write( "         3         x-486         969-2x          x-477         <2nd differences
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document.write( "           x-489         1455-3x         3x-1446               <3rd differences
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document.write( "                 1944-4x         6x-2901                       <4th differences
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document.write( "The row of 4th difference must be constant, so...

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document.write( "ANSWER: The missing number is 484.5

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document.write( "This result was confirmed with the matrix work I did on my TI-84 calculator.

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