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If(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + c_(3) x^(3) + …+ C_(n) x^(n), show that sum_(r=0)^(n) (C_(r) 3^(r+4))/((r+1)(r+2)(r+3)(r+4)) (1)/((n+1)(n+2)(n+3)(n+4))(4^(n+4) -sum_(t=0)^(3) ""^(n+4)C_(t^(3^(t)))). |
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Answer» SOLUTION :`LHS = sum_(r=0)^(n) (C_(r) 3^(r+4))/((r+1)(r+2)(r+3)(r+4))` ` =sum_(r=0)^(n)((C_(r)*3^(r+4))/((n+1)(n+2)(n+3)(n+4)))/(4!) 4!` `= sum_(r=0)^(n) (C_(r) 3^(r+4))/(""^(r+4)C_(4) *4!)= sum_(r=0)^(n)(n!)/(!(n-r)!) *(3^(r+4))/(((r+4)!)/(4!r!)*4!)` `=sum_(r=0)^(n) (n!*3^(r+4))/((n-r)!*(n+4)!)` `=sum_(r=0)^(n) (n!*3^(r+4))/((n-r)!*(n+4)!)*((n+1)(n+2)(n+3)(n+4))/((n+1)(n+2)(n+3)(n+4))` `=sum_(r=0)^(n) (n!*3^(r+4))/((n-r)!*(n+4)!(n+1)(n+2)(n+3)(n+4))` `=(1)/((n+1)(n+2)(n+3)(n+4))[sum_(r=0)^(n) ((n+4)!*3^(*r+4))/((n-r)!*(r+4)!)]` `=(1)/((n+1)(n+2)(n+3)(n+4))[sum_(r=0)^(n)""^(n+4)C_(r+4) 3^(r+4)]` `=(1)/((n+1)(n+2)(n+3)(n+4)){sum_(r=0)^(n)""^(n+4)C_(t)3^(t)}`[put r + 4 = t] `=(1)/((n+1)(n+2)(n+3)(n+4)){sum_(r=0)^(n)""^(n+4)C_(t)3^(t)- sum_(t=0)^(3) ""^(3^(t)n+4)C_(t)3^(t)} ` `=(1)/((n+1)(n+2)(n+3)(n+4)){(1+3)^(n+4) sum_(t=0)^(3) ""^(3^(t)n+4)C_(t)3^(t)}` `=(1)/((n+1)(n+2)(n+3)(n+4)){4^(n+4)- sum_(t=0)^(3) ""^(n+4)C_(t)3^(t)}` RHS . |
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