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If (1+x)n=C0+C1X+C2X2+C3X3.....CnXn

then the value of the following is:

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

6 years ago


Answers : (1)


Dear Jayesh


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)


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