Please help me with 13th sum
correct option is A

Dear Student,
Please find below the solution to the asked query:

S=3.nC0-8.nC1+13.nC2-18.nC3+..upto n+1 terms3,8,13 is an A.P. of first term and common difference 5rth term is given by:Tr=3+r-15=5r-2But we have to start from r=05r-2+5=5r+3From r=0 to r=n, there will be n+1 termsS=r=0n -1r 5r+3.nCr=5r=0n -1r r.nCr+3r=0n -1r .nCrS=5r=1n -1r n.n-1Cr-1+3r=0n -1r .nCr ....iAs r.nCr=n.n-1Cr-1We know that:1+xn=nC0+nC1x+nC2x2+....+ nCn xnPut x=-11-1n=nC0-nC1+nC2-....+-1n nCn0=r=0n -1r .nCri.e.r=0n -1r .nCr=0...ii1+xn-1=n-1C0+n-1C1x+n-1C2x2+...+n-1Cn-1xnPut x=-11-1n-1=n-1C0-n-1C1+n-1C2+...+-1n n-1Cn-11-1n-1=r=1n -1r n.n-1Cr-10=r=1n -1r n.n-1Cr-1r=1n -1r n.n-1Cr-1=0...iiiUsing ii and iii in i, we get:S=5×0+3×0=0OptionA

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