Prove that `(lfloor2n)/(lfloorn) = 1 xx 3 xx 5 xx …….xx (2n-1) xx 2^n`

In which of the following expressions,prime factorisation has been done: 56 = 1 xx 7 xx 2 xx 2 xx 2 .

Consider the following pattern: 2^3-1^3 = 1 +2 xx 1 xx 3 3 ^3 - 2 ^ 3 = 1 + 3 xx 2 xx 3 4 ^3 - 3 ^3 = 1 + 4 xx 3 xx 3 Using the above pattern,find the value of the following : 7^3 - 6^3 .

Consider the following pattern: 2^3-1^3 = 1 +2 xx 1 xx 3 3 ^3 - 2 ^ 3 = 1 + 3 xx 2 xx 3 4 ^3 - 3 ^3 = 1 + 4 xx 3 xx 3 Using the above pattern,find the value of the following : 12^3 - 11 ^ 3 .

Consider the following pattern: 2^3-1^3 = 1 +2 xx 1 xx 3 3 ^3 - 2 ^ 3 = 1 + 3 xx 2 xx 3 4 ^3 - 3 ^3 = 1 + 4 xx 3 xx 3 Using the above pattern,find the value of the following : 51 ^ 3 - 50 ^ 3 .

Consider the following pattern: 2^3-1^3 = 1 +2 xx 1 xx 3 3 ^3 - 2 ^ 3 = 1 + 3 xx 2 xx 3 4 ^3 - 3 ^3 = 1 + 4 xx 3 xx 3 Using the above pattern,find the value of the following : 20 ^ 3 - 19 ^ 3 .

Solve the following equation : 2^x xx 4^x=(8)^(1/3)xx(32)^(1/5) .

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) .