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\n" ); document.write( "Use mathematical induction....
\n" ); document.write( "(1) show it is true for n=1; then (2) show that if it is true for n=k then it is also true for n=k+1
\n" ); document.write( "(1) When n=1, the expression is 4(100)+9(10)+5 = 495 = 5(99).
\n" ); document.write( "The expression is divisible by 99 for n=1.
\n" ); document.write( "(2) Assume the expression is true for n=k and show that it follows that it is true for n=k+1:
\n" ); document.write( "Assume 4*10^(2k)+9*10(2k-1)+5 is divisible by 99. Then
\n" ); document.write( "4*10^(2(k+1))+9*10^(2(k+1)-1)+5 =
\n" ); document.write( "4*10^(2k+2)+9*10^(2k+1)+5 =
\n" ); document.write( "100(4*10^(2k)+9*10(2k-1)+5)-495
\n" ); document.write( "In that expression, (4*10^(2k)+9*10(2k-1)+5) is divisible by 99 and 495 is also divisible by 99, so we have shown that if the expression is true for n=k then it follows that it is true for n=k+1.
\n" ); document.write( "That completes the proof by mathematical induction.
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