If the coefficients of 4 consecutive terms in the expansion of `(1+x)^(n)` are `a_(1),a_(2),a_(3),a_(4)` respectively, then show that
`(a_(1))/(a_(1)+a_(2))+(a_(3))/(a_(3)+a_(4))=(2a_(2))/(a_(2)+a_(3))`

If a_(1), a_(2), a_(3), a_(4) are the coefficients of the 2^(nd), 3^(rd), 4^(th) and 5^(th) terms respectively in the binomial expansion of (1+x)^(n) where n is a positive integer prove that (a_(1))/(a_(1)+a_(2)) , (a_(2))/(a_(2)+a_(3)) , (a_(3))/(a_(3)+a_(4)) are in arithmetic progression.

If a_(1), a_(2),…a_(n) are in H.P. then a_(1).a_(2) + a_(2).a_(3) + a_(3).a_(4) + …+ a_(n-1) .a_(n) =

If (1+x+x^(2))^(n)=a_(0)+a_(1)x+a_(2)x^(2)+….+a_(2n)x^(2n) , then prove that a_(0)+a_(1)+a_(2)…..+a_(2n)=3^(n)

If the lines a_(1)x+b_(1)y=1,a_(2)x+b_(2)y=1,a_(3)x+b_(3)y=1, are concurrent then the points (a_(1),b_(1)),(a_(2),b_(2)),(a_(3),b_(3))

If a_(n) = sin^(n) theta + cos^(n) theta the P.T. (a_(3) -a_(5))/(a_(1))=(a_(5)-a_(7))/(a_(3))

If |(1+a_(1),a_(2),a_(3)),(a_(1),1+a_(2),a_(3)),(a_(1),a_(2),1+a_(3))|=0 then a_(1)+a_(2)+a_(3)=

If (1+3x-2x^(2))^(10)=a_(0)+a_(1)x+a_(2)x^(2).+…+a_(20)x^(20) then prove that a_(0)-a_(1)++a_(2)-a_(3)+……+a_(20)=4^(10)