# induction inequalities P3

• February 13th 2010, 03:40 AM
christina
induction inequalities P3
Prove using the principle of mathematical induction that

(1+x)^n> (or equal to) 1+nx, n=1,2,3...

thank you !
• February 13th 2010, 06:34 AM
Krizalid
• February 13th 2010, 05:37 PM
Quote:

Originally Posted by christina
Prove using the principle of mathematical induction that

(1+x)^n> (or equal to) 1+nx, n=1,2,3...

thank you !

This is very straightforward.

If $(1+x)^n\ \ge\ 1+nx$ ......(1)

then $(1+x)^{n+1}\ must\ be\ \ge\ 1+(n+1)x$ ....... (2)

Try to prove this using (1)
as this means (1) being true for some k causes (1) to be true for all n >k.

$(1+x)^k(1+x)\ \ge\ (1+kx)(1+x)$ ?

$(1+x)^{k+1}\ \ge\ 1+kx+x+kx^2$ ?

$(1+x)^{k+1}\ \ge\ 1+(k+1)x+kx^2$ which is > $1+(k+1)x$