Find n if `1^3 + 2^3 + 3^3+…………+n^3=2025`.

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

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 .

If 1+2+3+……………+n=45, then 1^3+2^3+3^3+…………+n^3 is:

If n is an odd integer greater than or equal to 1, then the value of n^3 - (n-1)^3 + (n-2)^3 - (n-3)^3 + .... + (-1)^(n-1) 1^3

Find n if : "^(2n)P_3=100 ^nP_2 .

Find n if "^(n-1)P_3 : "^nP_4 =1:9 .

Determine n if "^(2n)C_3 : "^nC_3 = 11 : 1

Determine n if "^(2n)C_3 : "^nC_3 =12:1

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