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find the sum of series nC1+(n+1)C2+(n+2)C3+.........+(n+r-1)Cr please could u help me how to proceed in these type of problems

find the sum of series
nC1+(n+1)C2+(n+2)C3+.........+(n+r-1)Cr
please could u help me how to proceed in these
type of problems

Grade:

4 Answers

Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago

Arun Kumar
IIT Delhi
Askiitians Faculty
Arun Kumar IIT Delhi
askIITians Faculty 256 Points
9 years ago
Little mistake in above answer

Saumya
13 Points
6 years ago
N(C1+C2+......Cr)+( (2-1)C2+(3-1)C3......) SUMMATION Cr + SUMMATION rCrSUMMATION Cr is 2^nAnd to solve rCr write expansion of (1+x)^n and differentiate and then put x=1
Kushagra Madhukar
askIITians Faculty 628 Points
3 years ago
Dear student,
Please find the solution to your problem.
 
nC1 + n+1C2 + n+2C3 + .........+ n+r–1Cr
= nC0 + nC1 + n+1C2 + n+2C3 + .........+ n+r-1Cr – nC0       {adding and subtracting nC0}
= n+1C1 + n+1C2 + n+2C3 + .........+ n+r-1Cr – 1
{Since we know, nCr + nCr+1 = n+1Cr+1}
= n+2C2 + n+2C3 + .........+ n+r-1Cr – 1
With each term we can apply the same formula
Hence, the condensed result will be
= n+rCr – 1 = n+rCn – 1
 
Thanks and regards,
Kushagra

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