Question

Using induction, verify the inequality. (1 + x)" ? 1 + nx, for x ?-1 and ? 1

Answer 1

Answer with explanation:

The Inequality should be:

   $(1+x)^n\geq 1+n x$

Where, n and x are any integers.

⇒For, x= -1

L HS

 $=[1+(-1)]^n\\\\=(0)^n\\\\=0$

R HS

→1+n × (-1)

=1-n

If, n is any Integer, then for, n=1

1-1=0

For, n=2

1-2= -1

....

So,   $(1+x)^n\geq 1+n x$

for, x=-1.

⇒For, x=1

L HS

 $=[1+(1)]^n\\\\=(2)^n$

For, n=1

L H S=1

For, =2

L H S=4

R HS

→1+n × (1)

=1+n

If, n is any Integer, then for, n=1

1+1=2

For, n=2

1+2= 3

....

So,    $(1+x)^n\geq 1+n x$

for, x=1.