If (1+x)ⁿ=C₀+C₁X+C₂X²+C₃X³.....C_nXⁿ

then the value of the following is:

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

Dear Jayesh

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

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

=∑ r(r+1)/2 ⁿC_r

= 1/2 ∑( r²+ r } ⁿC_r

 = 1/2 ∑( r(r-1) +r + r } ⁿC_r

 =1/2 ∑( r(r-1) +2r } ⁿC_r

=1/2 ∑( r(r-1)  ⁿC_r  + ∑  r  ⁿC_r    

=1/2 ∑( r(r-1) n(n-1)/r(r-1)  ^n-2C_r-2 + ∑  r  n/r ^n-1C_r-1   

 = 1/2 ∑(  n(n-1)  ^n-2C_r-2 + ∑    n ^n-1C_r-1   

= 1/2 n(n-1) ∑  ^n-2C_r-2 + n∑   ^n-1C_r-1

=   n(n-1)/2 *2^n-2 +n2^n-1

= n(n-1) 2^n-3 +n2^n-1

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

=n(n+3)2^n-3

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