If `A=[(3,-4),(1,-1)]`, prove by induction that `A_n=[(1+2n,-4n),(n,1-2n)],n in N`

If A=[(cos alpha,sinalpha),(-sinalpha,cosalpha)] , prove by induction that A^n=[(cosnalpha,sinnalpha),(-sinnalpha,cosnalpha)],n in N

If A=[(a,b),(0,1)] , where a ne 1 , then prove that A^(n) =[(a^(n), (b(a^(n)-1))/(a-1)),(0,1)], n in N .

Prove by the principle of mathematical induction that 4+8+12+….+4n=2n(n+1),AA n in N .

Prove by the principle of mathematical induction that 1+4+7+…..+(3n-2)=1/2n(3n-1),AA n in N .

For any natural number n prove by mathematical induction that: 1.2+2.3+3.4+…..+n(n+1)=1/3n(n+1)(n+2) .

Prove by the principle of mathematical induction that 2+5+8+11+….+(3n-1)=1/2n(3n+1),AA n in N .

Prove that: n(n-1)(n-2)…..(n-r+1)=(n!)/((n-r)!)

Prove by the principle of mathematical induction that 1+2+2^2+….2^(n-1)=2^n-1,AA n in N .

Find the value of .^(n-1)C_(n-1)

Prove that: ""^(n-1)P_r=(n-r)* ""^(n-1)P_(r-1)