Prove the following by the principle of mathematical induction: `a+(a+d)+(a+2d)++(a+(n-1)d)=n/2[2a+(n-1)d]`

Prove the following by using the Principle of mathematical induction AA n in N n<2^(n)

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

Prove the following by using the Principle of mathematical induction AA n in N 2^(n+1)>2n+1

Prove the following by the principle of mathematical induction: 2+5+8+11++(3n-1)=(1)/(2)n(3n+1)

Prove the following by using the Principle of mathematical induction AA n in N 3^(n)>2^(n)

Prove the following by the principle of mathematical induction: 1+2+2^(7)=2^(n+1)-1 for all n in N

Prove the following by using the Principle of mathematical induction AA n in N 2^(n+3)le(n+3)!

Prove the following by the principle of mathematical induction: (1)/(2)+(1)/(4)+(1)/(8)++(1)/(2^(n))=1-(1)/(2^(n))

Prove the following by the principle of mathematical induction: 1.2+2.2^(2)+3.2^(3)++n.2^(n)=(n-1)2^(n+1)+2

Prove the following by the principle of mathematical induction: 1^(2)+3^(2)+5^(2)++(2n-1)^(2)=(1)/(3)n(4n^(2)-1)