1. Find the sum
s_n =1+2x+3x^2 +4x^3 +...+nx^(n−1)
Hence,or otherwise, find the sum
Σ_(k=1) ^n k.2^k
2. Simplify the following
i. Σ_(r=0) ^n (_(2r−1) ^(2n) )
ii.Σ_(r=0) ^n (−1)^r r(_r ^n )
iii.Σ_(r=0) ^n (−1)^r (1/(r+1))(_r ^n )
iv.Σ_(r=0) ^n (_(2r) ^(2n) )
v.Σ_(r=0) ^n (−1)^r (_(n−r) ^(n+1) )
3.Find the sum
Σ_(r=0) ^(n−k) (_k ^(n−r) ), where k=0,1,2,3,...,n
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