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Question Number 53141 by gunawan last updated on 18/Jan/19

If n is an integer greater than unity, then the value of a−^n C_1 (a−1)+^n C_2 (a−2)−^n C_3 (a−3)+... ..+(−1)^n (a−n) is

Ifnisanintegergreaterthanunity,thenthevalueofa−nC1(a−1)+nC2(a−2)−nC3(a−3)+.....+(−1)n(a−n)is

Answered by tanmay.chaudhury50@gmail.com last updated on 18/Jan/19

a(1−nc_1 +nc_2 −nc_3 +...+(−1)^n nc_n )+(0+nc_1 −2nc_2 +3nc_3 −nc_4 +..−(−1)^n n) as_1 +s_2 (1+x)^n =1+nc_1 x+nc_2 x^2 +nc_3 x^3 +...+nc_n x^n put x=−1 0=1−nc_1 +nc_2 −nc_3 +..+(−1)^n nc_n so s_1 =0 n(1+x)^(n−1) =0+nc_1 +2nc_2 x+3nc_3 x^2 +...+n×nc_n x^(n−1) put x=−1 n×0=0+nc_1 −2nc_2 +3nc_3 −...+(−1)^n ×n×nc_n so s_2 =0 hence as_1 +s_2 =a×0+0=0

a(1−nc1+nc2−nc3+...+(−1)nncn)+(0+nc1−2nc2+3nc3−nc4+..−(−1)nn)as1+s2(1+x)n=1+nc1x+nc2x2+nc3x3+...+ncnxnputx=−10=1−nc1+nc2−nc3+..+(−1)nncnsos1=0n(1+x)n−1=0+nc1+2nc2x+3nc3x2+...+n×ncnxn−1putx=−1n×0=0+nc1−2nc2+3nc3−...+(−1)n×n×ncnsos2=0henceas1+s2=a×0+0=0

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