if `alpha, beta , ne 0 " and " f(n) =alpha^(n)+beta^(n)`
`" and " |{:(3,,1+f(1),,1+f(2)),(1+f(1),,1+f(2),,1+f(3)),(1+f(2),,1+f(3),,1+f(4)):}|`
`=k(1-alpha)^(2)(1-beta)^(2)(alpha-beta)^(2)` then k is equal to

If alpha,beta!=0 , and f(n)""=alpha^n+beta^n and {:|(3, 1+f(1),1+f(2)), (1+f(1),1+f(2),1+f(3)), (1+f(2),1+f(3),1+f(4))|:}=K(1-alpha)^2(1-beta)^2(alpha-beta)^2 , then K is equal to (1) alphabeta (2) 1/(alphabeta) (3) 1 (4) -1

If alpha,beta!=0 , and f(n)""=alpha^n+beta^n and |3 1+f(1)1+f(2)1+f(1)1+f(2)1+f(3)1+f(2)1+f(3)1+f(4)|=K(1-alpha)^2(1-beta)^2(alpha-beta)^2 , then K is equal to (1) alphabeta (2) 1/(alphabeta) (3) 1 (4) -1

If f(x) = alphax^(n) prove that alpha = (f'(1))/n

f(1)=1, n ge 1 f(n+1)=2f(n)+1 then f(n)=

If Delta = |(cos (alpha_(1) - beta_(1)),cos (alpha_(1) - beta_(2)),cos (alpha_(1) - beta_(3))),(cos (alpha_(2) - beta_(1)),cos (alpha_(2) - beta_(2)),cos (alpha_(2) - beta_(3))),(cos (alpha_(3) - beta_(1)),cos (alpha_(3) - beta_(2)),cos (alpha_(3) - beta_(3)))|" then " Delta equals

Let f(n)=2cosn xAAn in N , then f(1)f(n+1)-f(n) is equal to f(n+3) (b) f(n+2) f(n+1)f(2) (d) f(n+2)f(2)

If alpha and beta are roots of equation x^2 +px +q = 0 and f(n) = alpha^n+beta^n , then (i) f(n+1)+pf(n) -qf(n-1)=0 (ii) f(n+1)-pf(n) +qf(n-1)=0 (iii ) f(n+1)+pf(n) +qf(n-1)=0 (iv) f(n+1)-pf(n) -qf(n-1)=0

Let f(x) be a differentiable function and f(alpha)=f(beta)=0(alpha< beta), then the interval (alpha,beta)

Let f(n)=2cosn xAAn in N , then f(1)f(n+1)-f(n) is equal to (a) f(n+3) (b) f(n+2) (c) f(n+1)f(2) (d) f(n+2)f(2)

If alpha, beta be the roots of ax^(2)+bx+c=0(a, b, c in R), (c )/(a)lt 1 and b^(2)-4ac lt 0, f(n)= sum_(r=1)^(n)|alpha|^(r )+|beta|^(r ) , then lim_(n to oo)f(n) is equal to