`y^(2r+1)` in the expansion of `(y-1/y)^(2n+1)`

Let a_r denote the coefficient of y^(r-1) in the expansion of (1 + 2y)^(10) . If (a_(r+2))/(a_(r)) = 4 , then r is equal to

If k_r is the coefficient of y^(r - 1) in the expansion of (1 + 2y)^10 , in ascending powers of y , determine 'r' when (k_(r + 2))/(k_r) = 4

If k_r is the coefficient of y^(r-1) in the expansion of (1+2y)^10 , in ascending powers of y, determine r when (k_(r +2))/k_r=4 .

Show that the coefficient of x^(n)y^(n) in the expansion of {(1+x)(1+y)(x+y)}^(n) is C_(0)^(3)+C_(1)^(3) +C_(2)^(3) +…C_(n)^(3) .

Find the middle terms in the expansion of : (x/y+y/x)^(2n+1) .

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .

(1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + ...+ C_(n) x^(n) , show that sum_(r=0)^(n) C_(r)^(3) is equal to the coefficient of x^(n) y^(n) in the expansion of {(1+ x)(1 + y) (x + y)}^(n) .

The coefficient of x^(n) y^(n) in the expansion of [(1 + x)(1+y) (x +y)]^(n) , is