`underset(r=0)overset(n)sum(-1)^(r).a_(r )..^(n)C_(r )`
`= a_(0) xx .^(n)C_(0) - a_(1) xx .^(n)C_(1) + a_(2) xx .^(n)C_(2)-"….."+(-1)^(n)a_(n) xx .^(n)C_(0)`
`=` Coefficient of `x^(n)` in `(1-x+x^(2))^(n)(x-1)^(n)`
`=` Coefficient of `x^(n)` in `(x^(3)-1)^(n)`
`=...
Correct Answer - D
`(1-x)^(n)(1+x)^(n)=underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n)(1-x)^(n-r)`
or `(1-x+2x)^(n) = underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n-r)`
or `underset(r=0)overset(n)sum.^(n)C_(r)(1-x)^(n-r)(2x)^(r)=underset(r=0)overset(n)suma_(r)x^(r)(1-x)^(n-r)`
Comparing general term, we get `a_(r) = .^(n)C_(r)2^(r)`.
Correct Answer - C
In P, general term of the series is
`T_(r)=(.^(50-r)C_(r)(2r-1))/(.^(50)C_(r)(50+r))`
`=(.^(50+r)C_(r))/(.^(50)C_(r))(1-(50-r+1)/(50+r))`
`= (.^(50+r)C_(r))/(.^(50)C_(r))-(.^(50+r)C_(r))/(.^(50)C_(r))((50-r+1)/(50+r))`
Now,
`(.^(50+r)C_(r))/(.^(50)C_(r))((50-r+1)/(50+r))`
`= ((50-r+1)(50+r)!r!(50-r)!)/(r!50!(50+r)50!)`
`=((50-r+1)!(50+r-1)!)/(50!50!)`
`= (.^(50+r-1)C_(r-1))/(.^(50)C_(r-1))`
`rArr T(r) = (.^(50+r)C_(r))/(.^(50)C_(r)) - (.^(50+r-1)C_(r-1))/(.^(50)C_(r-1))= V(r) -...