Mathematical Induction - Inequalities (3 of 4: Starting with the assumption)

Increasing gaps trick kind of works here, too:

=2^(n+1)-2^n>3(n+1)²-3n²
=2×2^n-2^n>3(n²+2n+1)-3n²
=2^n>3n²+6n+3-3n²
=2^n>6n+3
But that last line can't be simplified further, so you need the additional explanation that 2^n will always grow by a factor of 2, while 6n+3 grows linearly, so the factor (compared to the previous n) will keep decreasing. Or you simply know that exponentials will always beat polynomials, but then you might as well skip the entire proof. :D

[shantanubanerjee]

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[sosoazl]

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[palmirinhawick]

Talk about Imaginary numbers, pleeease...

[Tyrantteemo]

Hi :)

I have a question. Why do you use k, k+1.. when (I guess) You can just go n, n+1. I dont see the point in exchanging n for k.
Thanks in advance.