Let `a ,b` be positive real numbers. If `a, A_1, A_2, b` be are in arithmetic progression `a ,G_1, G_2, b` are in geometric progression, and `a ,H_1, H_2, b` are in harmonic progression, show that `(G_1G_2)/(H_1H_2)=(A_1+A_2)/(H_1+H_2)`

If G_1 ,G_2 are geometric means , and A is the arithmetic mean between two positive no. then show that G_1^2/G_2+G_2^2/G_1=2A .

If H_1,H_2,H_3,.......H_n be n harmonic means between a and b then (H_1+a)/(H_1-a)+(H_n+b)/(H_n-b)=

If x_1 and x_2 be any two positive variables and if the arithmetic mean of these two values be A, geometric mean be G and harmonic mean be H, then peove that G^2 = AH

Let a_1 , a_2 ,... be in harmonic progression with a_1 = 5 and a_20 = 25 . The least positive integer n for which a_n lt 0

If two positive values of a variable be x_(1) and x_(2) and the arithmetic mean be A, geometric mean be G and the harmonic mean be H, then prove that G^(2)=AH

If the points ( a_1,b_1),(a_2,b_2) and (a_1+a_2,b_1+b_2) are collinear ,show that a_1b_2=a_2b_1 .

Let (1+x^2)^2(1+x)^n=sum_(k=0)^(n+4)a_k x^k . If a_1 , a_2 and a_3 are in arithmetic progression, then the possible value/values of n is/are a. 5 b. 4 c. 3 d. 2

If the coefficient of-four successive termsin the expansion of (1+x)^n be a_1 , a_2 , a_3 and a_4 respectivel. Show that a_1/(a_1+a_2)+a_3/(a_3+a_4)=2 a_2/(a_2+a_3)

If a_1,a_2,.....a_n are in H.P., then the expression a_1a_2 + a_2a_3 + ... + a_(n-1)a_n is equal to

Let A_r ,r=1,2,3, , be the points on the number line such that O A_1,O A_2,O A_3dot are in G P , where O is the origin, and the common ratio of the G P be a positive proper fraction. Let M , be the middle point of the line segment A_r A_(r+1.) Then the value of sum_(r=1)^ooO M_r is equal to (a) (O A_1(O S A_1-O A_2))/(2(O A_1+O A_2)) (b) (O A_1(O A_2+O A_1))/(2(O A_1-O A_2) (c) (O A_1)/(2(O A_1-O A_2)) (d) oo