∑n=0mCnn+ris equal to?
Cn+1n+m+1
Cnn+m+2
Cn-1n+m+3
None of these
Explanation for the correct answer:
Finding the value of the given expression:
Given that, ∑r=0mCnn+r
∑r=0mCnn+r=∑r=0mCrn+r=Cn0+Cn+11+Cn+22+..........+Cn+mm=n!0!n-0!+n+1!1!n+1-1!+Cn+22+..........+Cn+mm∵Cnr=n!r!n-r!=1+n+1+Cn+22+Cn+33+..........+Cn+mm=n+2+Cn+22+Cn+33+..........+Cn+mm=Cn+21+Cn+22+Cn+33+..........+Cn+mm∵Cn+r1=n+r!1!n+r-1!=n+r=C2n+3+C3n+3+...........Cn+mm∵Cnr-1+Cnr=Cn+1r=C2n+3+C3n+3+...........Cn+mm=C3n+4+C4n+4+...........Cn+mm∵Cnr-1+Cnr=Cn+1r
Continuing this process We get,
=Cn+mm-1+Cn+mm=Cn+m+1m∵Cnr-1+Cnr=Cn+1r=Cn+m+1n+1∵Cnr=Cnn-r
Hence, the correct answer is option (A).
If the sum of m term is equal to n and sum of its n terms is equal to m then prove that sum of(m+n) terms is equal to -(m+n)
If sum of n term is equal to n and sum of n term is equal to m
Find the sum of (m+n).