induction inequalities P1

**christina** · February 13th 2010, 03:38 AM

Prove using the principle of mathematical induction that

3^n > 4n^2 +5n for n=4,5...

thank you!

**Krizalid** · February 13th 2010, 06:33 AM

show your progress.

**Archie Meade** · February 15th 2010, 03:14 AM

Originally Posted by christina

Prove using the principle of mathematical induction that

3^n > 4n^2 +5n for n=4,5...

thank you!

This is very straightforward.

$3^n>4n^2+5n$ ?

If this is true, we try to use it to prove if

$3^{n+1}>4(n+1)^2+5(n+1)$

$(3)3^n>4(n^2+2n+1)+5(n+1)$ ?

$3^n+(2)3^n>4n^2+4(2n)+4+5n+5$ ?

$3^n+[(2)3^n]>4n^2+5n+[4(2n)+9]$

Therefore, we only need ask if

$2(3^n)>4(2n)+9,\ n\ \ge\ 4$

$3^n+3^n>4n+4.5+4n+4.5$

As $n^2>n$ and $5n>4.5$ , for n=4 and above,

only the first term needs to be tested.

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