Prove the following by using the principle of mathematical induction for all `n in N`:`1/2+1/4+1/8+dotdotdot+1/(2^n)=1-1/(2^n)`

Prove the following by using the principle of mathematical induction for all n in N : (2n+7)<(n+3)^2 .

Prove the following by using the principle of mathematical induction for all n in N : 1^2+3^2+5^2+dotdotdot+(2n-1)^2=(n(2n-1)(2n+1))/3

Prove the following by using the principle of mathematical induction for all n in N : 1 + 2 + 3 + ... + n <1/8(2n+1)^2 .

Prove the following by using the principle of mathematical induction for all n in N : 10^(2n-1)+1 is divisible by 11.

Prove the following by using the principle of mathematical induction for all n in N : 1/(2. 5)+1/(5. 8)+1/(8. 11)+...+1/((3n-1)(3n+2))=n/((6n+4))

Prove the following by using the principle of mathematical induction for all n in N : 1/(3. 5)+1/(5. 7)+1/(7. 9)+...+1/((2n+1)(2n+3))=n/(3(2n+3)) .

Prove the following by using the principle of mathematical induction for all n in N : (1+3/1)(1+5/4)(1+7/9)...(1+((2n+1))/(n^2))=(n+1)^2

Prove the following by using the principle of mathematical induction for all n in N : (1+1/1)(1+1/2)(1+1/3)...(1+1/n)=(n+1)

Prove the following by using the principle of mathematical induction for all n in N : 1/(1. 2. 3)+1/(2. 3. 4)+1/(3. 4. 5)+...+1/(n(n+1)(n+2))=(n(n+3))/(4(n+1)(n+2))

Prove the following by using the principle of mathematical induction for all n in N : 1^3+2^3+3^3+""dot""""dot""""dot+n^3=((n(n+1))/2)^2