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# If (1+x)n=C0+C1X+C2X2+C3X3.....CnXnthen the value of the following is:(1)C1+(1+2)C2+(1+2+3)C3........(1+2+3.....n)Cn= ?

147 Points
11 years ago

Dear Jayesh

(1)C1+(1+2)C2+(1+2+3)C3........(1+2+3.....n)Cn

or ∑ (1+2+3...+r) nCr      where r varry from 1 to n

=∑ r(r+1)/2 nCr

= 1/2 ∑( r2+ r } nCr

= 1/2 ∑( r(r-1) +r + r } nCr

=1/2 ∑( r(r-1) +2r } nCr

=1/2 ∑( r(r-1)  nC+ ∑  r  nCr

=1/2 ∑( r(r-1) n(n-1)/r(r-1)  n-2Cr-2 + ∑  r  n/r n-1Cr-1

= 1/2 ∑(  n(n-1)  n-2Cr-2 + ∑    n n-1Cr-1

= 1/2 n(n-1) ∑  n-2Cr-2 + n∑   n-1Cr-1

=   n(n-1)/2 *2n-2 +n2n-1

= n(n-1) 2n-3 +n2n-1

=n 2n-3(n-1 +4)

=n(n+3)2n-3

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