If f(x) = 10^(2x+1) and x is a non-negative integer, prove that f(x+2) - f(x) is a divisible by 99.

Answer 1

If f(x) = 10^(2x+1), then f(x + 2) = 10^(2(x+2)+1) = 10^(2x+5)

So f(x + 2) - f(x) = 10^{(2x + 5)} - 10^(2x+1)

But 10^(2x+5) = 10^2x • 10⁵ and 10^(2x+1) = 10^2x • 10¹ so:

f(x + 2) - f(x) = 10^2x • 10⁵ - 10^2x • 10¹ = 10^2x(10⁵ - 10¹) = 10^2x(100,000 - 10) = 10^2x(99990) = 10^2x(99 • 1010)

Since 99 is a factor of f(x + 2) - f(x), then it is divisible by 99