Prove thatnC0+ 2 nC1+ 3 nC2+ +(n+1) nCn= - Maths - Binomial Theorem

Question

Prove thatnC0+ 2 nC1+ 3
nC2+ +(n+1) nCn= (n+2)
2( n-1)

Ishwarmani · Accepted Answer

Dear Student, Please find below the solution to the asked query:    C0n+2·C1n+3·C2n+
+n+1·Cnn Its rth term is tr = r·Cr-1n                       =r-1+1·Cr-1n                       =r-1·Cr-1n+1·Cr-1n                       =n·Cr-2n-1+Cr-1n                          as r-1·Cr-1n=n
Cr-2n-1 C0+2·C1n+3·C2n+
+n+1·Cnn = ∑r = 1n+1tr⇒ ∑r = 1n+1tr =∑r = 1n+1 n·Cr-2n-1+∑r = 1n+1Cr-1n                 =n∑r = 1n+1 Cr-2n-1+∑r = 1n+1Cr-1n                   =nC0n-1+C1n-1+
+Cn-1n-1+C0n+C1n+
+Cnn                 =n·2n-1+2n                 =n+22n-1Hope this information will clear your doubts about the topic
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Prove thatnC0+ 2.nC1+ 3.nC2+ ........... +(n+1).nCn= (n+2).2( n-1).

Prove thatⁿC₀+ 2.ⁿC₁+ 3.ⁿC₂+ ........... +(n+1).ⁿC_n= (n+2).2⁽ ^n-1).