document.write( "Question 1209928: The function f satisfies\r
\n" ); document.write( "\n" ); document.write( "f(a + b) = f(a) + f(b) - ab
\n" ); document.write( "for all nonnegative integers a and b, and f(1) = 7. Compute f(123).
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Algebra.Com's Answer #851076 by greenestamps(13200)\"\" \"About 
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\n" ); document.write( "Given:
\n" ); document.write( "f(1)=7
\n" ); document.write( "f(a+b)=f(a)+f(b)-ab

\n" ); document.write( "To find: f(123)

\n" ); document.write( "Find f(2):
\n" ); document.write( "f(2)=f(1+1)=f(1)+f(1)-1*1=7+7-1=13

\n" ); document.write( "Find f(3):
\n" ); document.write( "f(3)=f(1+2)=f(1)+f(2)-2=7+13-2=18

\n" ); document.write( "Find f(4):
\n" ); document.write( "f(4)=f(1+3)=f(1)+f(3)-3=7+18-3=22

\n" ); document.write( "Find f(4) in a different way to make sure the recursive definition is valid:
\n" ); document.write( "f(4)=f(2+2)=f(2)+f(2)-4=13+13-4=22

\n" ); document.write( "Find f(5) in two different ways:
\n" ); document.write( "f(5)=f(1+4)=f(1)+f(4)-4=7+22-4=25
\n" ); document.write( "f(5)=f(2+3)=f(2)+f(3)-6=13+18-6=25

\n" ); document.write( "The recursive definition appears to be valid.

\n" ); document.write( "The values of f(1) to f(5) form a sequence with a clear pattern:

\n" ); document.write( "7, 13, 18, 22, 25, ...

\n" ); document.write( "The differences between successive terms are decreasing by 1.

\n" ); document.write( "To find the value of f(123), we want to have an explicit formula for the n-th term. One way we can find that formula is using the method of finite differences.

\n" ); document.write( "Here is a display of the first few terms of the sequence and the first and second differences:
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document.write( "   7   13   18   22   26\r\n" );
document.write( "     6    5    4    3\r\n" );
document.write( "       -1   -1   -1

\n" ); document.write( "The constant difference of 1- means the sequence can be produced with a polynomial of degree 2 with leading coefficient -1/(2!) = -1/2. So the sequence can be formed with a polynomial of the form

\n" ); document.write( "\"t%28n%29=%28-1%2F2%29n%5E2%2Ban%2Bb\"

\n" ); document.write( "To find the coefficients a and b, we can compare the given sequence to the sequence formed by the polynomial \"%28-1%2F2%29n%5E2\".
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document.write( "\r\n" );
document.write( "   t(n) (-1/2)n^2  an+b\r\n" );
document.write( "  ----------------------\r\n" );
document.write( "    7     -1/2     15/2\r\n" );
document.write( "   13      -2      15 = 30/2\r\n" );
document.write( "   18     -9/2     45/2\r\n" );
document.write( "   22      -8      30 = 60/2\r\n" );
document.write( "   ...

\n" ); document.write( "We can see that the differences are just (15/2)n, so the explicit formula for the n-th term is

\n" ); document.write( "\"t%28n%29=%28-1%2F2%29n%5E2%2B%2815%2F2%29n\"
\n" ); document.write( "\"t%28n%29=%2815n-n%5E2%29%2F2\"
\n" ); document.write( "\"t%28n%29=%28n%2815-n%29%29%2F2\"

\n" ); document.write( "So

\n" ); document.write( "ANSWER: \"t%28123%29=%28123%2815-123%29%29%2F2=%28%28123%29%28-108%29%29%2F2=-6642\"

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