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Question

C0+32.C1+93.C2+274.C3+...........+3nn+1.Cn=4n+113(n+1).Is it true ?If true enter 1 else 0.

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Solution

Solve: we have Given.

c0+32c1+93c2+274c3++3nn+1cn=4n+113(n+1)

By binomial expansion

(1+x)n=nc0+nc1x+nc2x2+..ncnxn

on integrating both sides with respect to x we get

x0(1+x)ndx=(nc0+nc1x+nc2x2+ncnxn)dx

(1+x)n+11n+1=nc0x+nc1x22+nc2x33+ncnxn+1n+1

on putting x=3 we get

4n+11n+1=3nc0+92nc1+273nc2+..3n+1n+1ncn

4n+11n+1=3[nc0+32nc1+93nc2+3n+1n+1ncn]

nc0+32nc1+93nc2+3nn+1=4n+113(n+1)

Hence It is correct
so, correct answer is 1 .


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