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1)

If  αR,nN  and n+2(n-1)+3(n-2)+....(n-1)2+n.1= α n(n+1)(n+2), then α=  


A) 12

B) 13

C) 15

D) 16

Answer:

Option D

Explanation:

We have,

 Tr=r[n(r1)]

= r(n-r+1)

= nrr2+r

      nr=1Tr=nr=1(nrr2+r)

=nr=1[(n+1)rr2]=(n+1)nr=1rnr=1r2

  =(n+1)n(n+1)2n(n+1)(2n+1)6

  =n(n+1)2[(n+1)(2n+13)]

=n(n+1)2[3n+32n13]

  =n(n+1)2[n+23]

=n(n+1)(n+2)6

      α  = 16