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Prove that sum(r=1)^5 ^5Cr=31...

Prove that ` sum_(r=1)^5 ^5C_r=31 `

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Prove that sum_(r=0)^(5) ""^(5)C_(r )=32 .

If D_r=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1| then prove that sum_(r=1)^n D_r=0.

If D_r=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1| then prove that sum_(r=1)^n D_r=0.

Prove that sum_(r = 1)^n r^3 ((C_r)/(C_(r - 1))^2) = (n (n + 1)^2 (n+2))/(12)

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

Prove that sum_(r = 0)^n r^2 . C_r = n (n +1).2^(n-2)

Prove that sum_(r = 1)^(n+1) (2^(r +1) C_(r - 1) )/(r (r + 1)) = (3^(n+2) - 2n - 5)/((n+1)(n+2))

Prove that sum_(r = 1)^(n+1) (2^(r +1) C_(r - 1) )/(r (r + 1)) = (3^(n+2) - 2n - 5)/((n+1)(n+2))

Prove that sum_(r=1)^(10) = sin^(2) rpi/18 =5 .

Prove that sum_(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)(.^n C_r)=1/n .