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Question

If(1+x)^n =C0+C1x+C2x^2+......Cnx^n, prove that

1) 3C0-8C1+13C2-18C3+.....upto (n+1) terms

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Solution

sigma (r=0 to n) (-1)r(5r+3)nCr
=5 sigma(r=0 to n)(-1)rrnCr+3sigma(r=0ton)(-1)rnCr
=5sigma(r=0 to n)(-1)rr.(n/r)n-1Cr-1+3sigma(r=0ton)(-1)rnCr
=5nsigma(r=0 to n) (-1)r n-1Cr-1+3sigma(r=0ton)(-1)rnCr
=5n[ -n-1C0+n-1C1n-1C2+.........(-1)n n-1Cn-1]+ 3 [nC0-nC1+....+(-1)nnCn]
=5n(0) +3(0)
=0+0
=0

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