ਸਾਰੇ `nge1` ਲਈ, ਸਿੱਧ ਕਰੋ ਕਿ `1^2+ 2^2 + 3^2+ 4^2+. . . + n^2 =(n(n+1)(2n+1))/6`

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

Prove the following by using the principle of mathematical induction for all n in N :- 1.3 + 2.3^2 + 3.3^3 +... + n.3^n =((2n-1)3^(n+1) +3)/4 .

Prove the following by the principle of mathematical induction: 1^2+2^2+3^2++n^2=(n(n+1)(2n+1))/6

By the Principle of Mathematical Induction, prove the following for all n in N : 1.3+2.3^2 +3.3^3+.....+ n.3^n= ((2n-1)3^(n+1)+3)/4 .

Show that : (1 xx 2^2+2 xx 3^2+. .+n xx(n+1)^2)/(1^2 xx 2+2^2 xx 3+. .+n^2 (n+1)) =(3 n+5)/(3 n+1) .

Prove the following by using the principle of mathematical induction for all n in N :- 1.2 + 2.3 + 3.4 +... +n.(n+1)=[(n(n+1)(n+2))/3]

Prove the following by using the principle of mathematical induction for all n in N :- 1.2.3 + 2.3.4 +...+n(n+1)(n+2)=(n(n+1)(n+2)(n+3))/4 .

Find the value of (3^n × 3^(2n + 1))/(3^(2n) × 3^(n - 1))

The sum to infinity of the series 1+2(1-(1)/(n))+3(1-(1)/(n))^(2)+ . . .. . . . , is (A) n^(2) (B) n(n+1) (C) n(1+(1)/(n))^(2) (D)None of these