If `a,A_1,A_2,` b are in A.P., prove that `A_1 + A_2 = a + b`

If A_1 and A_2 are two A.M.’s between a and b, prove that : A_1+A_2 =a+b .

If a ,a_1, a_2, a_3, a_(2n),b are in A.P. and a ,g_1,g_2,g_3, ,g_(2n),b . are in G.P. and h s the H.M. of aa n db , then prove that (a_1+a_(2n))/(g_1g_(2n))+(a_2+a_(2n-1))/(g_1g_(2n-1))++(a_n+a_(n+1))/(g_ng_(n+1))=(2n)/h

If A_1 and A_2 are two A.M.’s between a and b, prove that : (2A_1-A_2)(2A_2-A_1) =ab .

If a_1,a_2,a_3,……………….. are in A.P., then the value of: {:|(a_1,a_2,1),(a_2,a_3,1),(a_3,a_4,1)| is equal to

If the sequence a_1, a_2, a_3,....... a_n ,dot forms an A.P., then prove that a_1^2-a_2^2+a_3^2-a_4^2+.......+ a_(2n-1)^2 - a_(2n)^2=n/(2n-1)(a_1^2-a_(2n)^2)

If a_1, a_2, a_3,.....a_n are in H.P. and a_1 a_2+a_2 a_3+a_3 a_4+.......a_(n-1) a_n=ka_1 a_n , then k is equal to

If a_1, a_2 , ............, a_n are in A.P. and a_i >0 for all i , prove that : 1/(sqrt a_1+sqrt a_2)+1/(sqrt a_2+sqrt a_3)+.............+ 1/(sqrt (a_(n-1))+sqrt a_n)= (n-1)/(sqrt a_1+sqrt a_n) .

If the oints (a_1b_1),(a_2b_2) and (a_1-a_2,b_1-b_2) are collinear, then show that a_1b_2=a_2b_1

Let a_1, a_2, ,a_(10) be in A.P. and h_1, h_2, h_(10) be in H.P. If a_1=h_1=2a n da_(10)=h_(10)=3,t h e na_4h_7 is 2 b. 3 c. 5 d. 6

If A_1, A_2, …, A_n are n independent events, such that P(A_i)=(1)/(i+1), i=1, 2,…, n , then the probability that none of A_1, A_2, …, A_n occur, is