1) If α∈R,n∈N 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 DExplanation:We have, Tr=r[n−(r−1)] = r(n-r+1) = nr−r2+r ∴ ∑nr=1Tr=∑nr=1(nr−r2+r) =∑nr=1[(n+1)r−r2]=(n+1)∑nr=1r−∑nr=1r2 =(n+1)n(n+1)2−n(n+1)(2n+1)6 =n(n+1)2[(n+1)−(2n+13)] =n(n+1)2[3n+3−2n−13] =n(n+1)2[n+23] =n(n+1)(n+2)6 ∴ α = 16