Prove that: . n`^C_0+2.^nC_1+…2^n.^nC_n=3^n` for every natural number n.

Prove that 1+2+2^(2)+ . . .+2^(n)=2^(n+1)-1 , for all natural number n.

Prove that C_0 + 2.C_1 + 4.C_2 + 8.C_3 + ……+2^n.C_n = 3^n

Show that n^3+(n+1)^3+(n+2)^3 is divisible by 9 for every natural number n .

Prove that 3.C_0 + 6.C_1 + 12.C_2 + ………+3.2^n.C_n = 3^(n+1)

prove that 1+5+9+ . . .+(4n-3)=n(2n-1), for all natural number n.

if (1+a)^(n)=.^(n)C_(0)+.^(n)C_(1)a++.^(n)C_(2)a^(2)+ . . .+.^(n)C_(n)a^(n) , then prove that (.^(n)C_(1))/(.^(n)C_(0))+(2(.^(n)C_(2)))/(.^(n)C_(1))+(3(.^(n)C_(3)))/(.^(n)C_(2))+. . . +(n(.^(n)C_(n)))/(.^(n)C_(n-1))= Sum of first n natural numbers.

Prove that log_n(n+1)>log_(n+1)(n+2) for any natural number n > 1.

Prove that .^n C_0 . ^(2n) C_n- ^n C_1 . ^(2n-2)C_n+^n C_2 . ^(2n-4)C_n-=2^ndot

Prove that (n !+1) is not divisible by any natural number between 2 and n

Prove that (n !+1) is not divisible by any natural number between 2 and n