`A_(1), A_(2), …, A_(n)` are independent events such that `P(A_(i))=1-q_(i), i=1,2, …, n`. Prove that,
`P(A_(1) cup A_(2) cup … cup A_(n))=1-q_(1)q_(2)…q_(n)`.

If A_(1), A_(2),..,A_(n) are any n events, then

A_1, A_2……….A_n are n independent events such that P(A_i) = 1 - q_i, I = 1, 2, 3…………….n Prove that P[overset(n)underset(i)uuu A_i] = 1 - q_1q_2……….q_n

A_(1), A_(2) and A_(3) are three events. Show that the simultaneous occurrence of the events is P(A_(1) cap A_(2) cap A_(3))=P(A_(1))P(A_(2)//A_(1))P[A_(3)//(A_(1) cap A_(2))] State under which condition P(A_(1) cap A_(2) cap A_(3))=P(A_(1))P(A_(2))P(A_(3))

If a_(1) ge 0 for all t and a_(1), a_(2), a_(3),….,a_(n) are in A.P. then show that, (1)/(a_(1)a_(3)) + (1)/(a_(3)a_(5)) +...+ (1)/(a_(2n-1).a_(2n+1)) = (n)/(a_(1)a_(2n+1))

If a_(1),a_(2),a_(3)anda_(4) be the coefficients of four consecutive terms in the expansion of (1+x)^(n) , then prove that (a_(1))/(a_(1)+a_(2)),(a_(2))/(a_(2)+a_(3))and(a_(3))/(a_(3)+a_(4)) are in A.P.

If a_(1), a_(2), a_(3),…, a_(2k) are in A.P., prove that a_(1)^(2) - a_(2)^(2) + a_(3)^(2) - a_(4)^(2) +…+a_(2k-1)^(2) - a_(2k)^(2) = (k)/(2k-1)(a_(1)^(2) - a_(2k)^(2)) .

If positive integers a_(1), a_(2), a_(3),… are ion A.P. such that a_(8) +a_(10) =24, then the value of a_(9) is-

If a_(1), a_(2), a_(3), …., a_(n) are in H.P., 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)

If a_(1), a_(2),…,a_(n) are in G.P., then show that (1)/(a_(1)^(2) - a_(2)^(2)) + (1)/(a_(2)^(2) - a_(3)^(2))+...+ (1)/(a_(n-1)^(2) - a_(n)^(2)) = (r^(2))/((1-r^(2))^(2))[(1)/(a_(n)^(2))-(1)/(a_(1)^(2))]

Let a_(1), a_(2),…,a_(n) be fixed real numbers and define a function f(x)=(x-a_(1))(x-a_(2))…(x-a_(n)). What is lim_(xrarra_(1))f(x) ? For some a ne a_(1), a_(2), …..,a_(n) , compute lim_(xrarra)(f(x) .