Second principal of mathematical induction

On principle 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.

Prove by using the principal of mathematical inductions 4+ 44 +444+ ….+ 444…4 ( n digits ) = ( 4)/(81) (10 ^(n+1) - 9n -10)

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.

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 n in NN .

If ninNN , then by principle of mathematical induction prove that, 10^(n)+3*4^(n+2)+5" "[nge0] is divisible by 9.

By the Principal of mathematical induction, prove that 1+5+5^2+…..+5^(n-1)=(5^n-1)/4

State the principle of mathematical induction.

Using the principal of Mathematical induction, prove that 1/2+1/4+1/8+…….+1/2^n=1-1/2^n