Question 1179135
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Use mathematical induction....<br>
(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<br>
(1) When n=1, the expression is 4(100)+9(10)+5 = 495 = 5(99).<br>
The expression is divisible by 99 for n=1.<br>
(2) Assume the expression is true for n=k and show that it follows that it is true for n=k+1:<br>
Assume 4*10^(2k)+9*10(2k-1)+5 is divisible by 99.  Then<br>
4*10^(2(k+1))+9*10^(2(k+1)-1)+5 =
4*10^(2k+2)+9*10^(2k+1)+5 =
100(4*10^(2k)+9*10(2k-1)+5)-495<br>
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.<br>
That completes the proof by mathematical induction.<br>