document.write( "Question 1163528: please help me solve this problem\r
\n" ); document.write( "\n" ); document.write( "P(x) function polynomial 2nd degree P(1)=1 P(2)=7 P(3)=19\r
\n" ); document.write( "\n" ); document.write( "we have natural integer n ⁄ n≥1\r
\n" ); document.write( "\n" ); document.write( "proof that P(1)+P(2)+...+P(n)=n^3\r
\n" ); document.write( "\n" ); document.write( "please help me solve this problem \n" ); document.write( "

Algebra.Com's Answer #787655 by greenestamps(13198) $\"\"$ $\"About$

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\n" ); document.write( "1. Finding the quadratic function from the given data points....

\n" ); document.write( "A traditional algebraic method....

\n" ); document.write( "The general quadratic polynomial is $\"an%5E2%2Bbn%2Bc\"$ .

\n" ); document.write( "Use the three given data points to form three equations in a, b, and c; then solve the system of equations.

\n" ); document.write( " $\"a%2Bb%2Bc+=+1\"$
\n" ); document.write( " $\"4a%2B2b%2Bc+=+7\"$
\n" ); document.write( " $\"9a%2B3b%2Bc+=+19\"$

\n" ); document.write( " $\"3a%2Bb+=+6\"$
\n" ); document.write( " $\"5a%2Bb+=+12\"$

\n" ); document.write( " $\"2a+=+6\"$
\n" ); document.write( " $\"a+=+3\"$

\n" ); document.write( " $\"9%2Bb+=+6\"$
\n" ); document.write( " $\"b+=+-3\"$

\n" ); document.write( " $\"3%2B%28-3%29%2Bc+=+1\"$
\n" ); document.write( " $\"c+=+1\"$

\n" ); document.write( "ANSWER: The quadratic function for the given points is $\"3n%5E2-3n%2B1\"$

\n" ); document.write( "Here is another method for finding the quadratic function -- far less common than the preceding method; but useful if you know how to use it.

\n" ); document.write( "In the quadratic function $\"an%5E2%2Bbn%2Bc\"$ , there is a common second difference of 2a.

\n" ); document.write( "Use that to determine coefficient a.

\r\n" );
document.write( "    1   7   19   <-- given sequence\r\n" );
document.write( "      6   12     <-- first differences (differences between successive terms)\r\n" );
document.write( "        6        <-- second difference (difference between successive first differences)

\n" ); document.write( "The second difference is 6; since 2a=6, that means a=3.

\n" ); document.write( "Now compare the whole function $\"an%5E2%2Bbn%2Bc\"$ to $\"an%5E2\"$ to determine coefficients b and c. The difference between those two functions is the linear polynomial $\"bx%2Bc\"$ .

\n" ); document.write( "n=1: an^2 = 3(1^2) = 3; P(1) = 1; difference is -2
\n" ); document.write( "n=2: an^2 = 3(2^2) = 12; P(2) = 7; difference is -5
\n" ); document.write( "n=3: an^2 = 3(3^2) = 27; P(3) = 19; difference is -8

\n" ); document.write( "The linear polynomial $\"bx%2Bc\"$ that produces the sequence -2, -5, -8 is $\"-3n%2B1\"$ .

\n" ); document.write( "Therefore, the polynomial that produces the given sequence is $\"%28an%5E2%29\"$ + $\"%28bn%2Bc%29\"$ = $\"%283n%5E2%29\"$ + $\"%28-3n%2B1%29\"$ = $\"3n%5E2-3n%2B1\"$ .

\n" ); document.write( "2. Proving that P(1)+P(2)+...+P(n)=n^3....

\n" ); document.write( "Prove using mathematical induction. To do that proof, we need to show two things:

\n" ); document.write( "(a) P(1) = 1:
\n" ); document.write( " $\"P%281%29+=+3%281%5E2%29-3%281%29%2B1+=+3-3%2B1+=+1\"$ OK

\n" ); document.write( "(b) The difference between $\"n%5E3\"$ and $\"%28n-1%29%5E3\"$ is P(n).

\n" ); document.write( " $\"n%5E3-%28n-1%29%5E3+=+n%5E3-%28n%5E3-3n%5E2%2B3n-1%29+=+3n%5E2-3n%2B1\"$ OK

\n" ); document.write( "The proof is complete.

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

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