If ` a_(0), a_(1), a_(2),…` are the coefficients in the expansion of
` (1 + x + x^(2))^(n)` in ascending powers of x, prove that
`a_(0) a_(2) - a_(1) a_(3) + a_(2) a_(4) - …+ a_(2n-2) a_(2n)= a_(n+1)` .

If a_(0), a_(1),a_(2),… are the coefficients in the expansion of (1 + x + x^(2))^(n) in ascending powers of x, prove that if E_(1) = a_(0) + a_(3) + a_(6) + …, E_(2) = a_(1) + a_(4) + a_(7) + … and E_(3) = a_(2) + a_(5) + a_(8) + ..." then " E_(1) = E_(2) = E_(3) = 3^(n-1)

( 1 + x + x^(2))^(n) = a_(0) + a_(1) x + a_(2) x^(2) + …+ a_(2n) x^(2n) , then a_(0) + a_(1) + a_(2) + a_(3) - a_(4) + … a_(2n) = .

If the expansion in power of x of the function (1)/(( 1 - ax)(1 - bx)) is a_(0) + a_(1) x + a_(2) x^(2) + a_(3) x^(3) + …, then a_(n) is

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

Find a_(1) and a_(9) if a_(n) is given by a_(n) = (n^(2))/(n+1)

It a_(1) , a_(2) , a_(3) a_(4) be in G.P. then prove that (a_(2)-a_(3))^(2) + (a_(3) - a_(1))^(2) + (a_(4) -a_(2))^(2) = (a_(1)-a_(4))^(2)

If a_(0) = 0.4 and a_(n+1) = 2|a_(n)|-1 , then a_(5) =

If a_(1),a_(2),a_(3),a_(4),,……, a_(n-1),a_(n) " are distinct non-zero real numbers such that " (a_(1)^(2) + a_(2)^(2) + a_(3)^(2) + …..+ a_(n-1)^(2))x^2 + 2 (a_(1)a_(2) + a_(2)a_(3) + a_(3)a_(4) + ……+ a_(n-1) a_(n))x + (a_(2)^(2) +a_(3)^(2) + a_(4)^(2) +......+ a_(n)^(2)) le 0 " then " a_(1), a_(2), a_(3) ,....., a_(n-1), a_(n) are in

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) = 2 and a_(n) - a_(n-1) = 2n (n ge 2) , find the value of a_(1) + a_(2) + a_(3)+…+a_(20) .