Second principal of mathematical induction

If n gt 1 and x gt -1, x ne 0 , show by the principal of mathematical induction (1+x)^(n) gt 1 +n x.

If A=[[costheta,isintheta],[isintheta,costheta]], then prove by principal of mathematical induction that A^n=[[cosntheta, i sinn theta],[isin n theta, cos n theta]] for all nin NN.

Using principle of mathematical induction prove that sqrt(n) =2

Prove by the principle of mathematical induction that n<2^(n) for alln in N

Prove the rule of exponents (ab)^(n)=a^(n)b^(n) by using principle of mathematical induction for every natural number.

Prove by using principle of mathematical induction :2^(n)<3^(n),n in N

Using the principle of mathematical induction to show that tan^(-1)(n+1)x-tan^(-1)x , forall x in N .

Use the principle of mathematical induction to show that a^(n) - b^9n) is divisble by a-b for all natural numbers n.

Using the principle of mathematical induction, prove that (7^(n)-3^(n)) is divisible by 4 for all n in N .

Prove the following by the principle of mathematical induction: 1+3+3^(2)++3^(n-1)=(3^(n)-1)/(2)