Show the following identities for nonnegative integers k n s

Show the following identities for non-negative integers k, n sigma_k=0^n (n k) = 2^n For 0 lessthanorequalto k lessthanorequalto n; (n+1 k+1) = (n k+1) + (n k)

Solution

Solution : 1. we know the Binomal Expansion :

( 1 + x ) ⁿ = nc_{0 +} nc₁ x + nc₂ x²  + ....... + nc_n xⁿ . put x = 1

(1+1)ⁿ = nc₀ + nc₁+ nc₂ + ------ nc_n

2ⁿ = nc₀+ n c₁ + nc₂  ----+ nc_{n ------(1)}

2. RHS ; n c _(k+1) + n c _{k = n ! / ( n - k - 1) ! ( k+1) ! + n ! / ( n - k ) ! K ! = [ n ! / k ! ( n - k - 1) ! { 1 / (k+1) + 1 / ( n -k ) } ]}

   _{= [ n ! / k! ( n - k - 1) ! { (n - k + k +1) / (n-k) (k+1)}]}

   _{= n ! (n+1) / k ! (k+1) ( n-k-1) ! ( n- k)}

   = (n+1) ! / ( k+ 1) ! ( n-k ) ! = (n+1) c _{(k+1) =LHS}