Home
Class 12
MATHS
If A(1), A(2), A(3) , G(1), G(2), G(3) ,...

If `A_(1)`, `A_(2)`, `A_(3)` , `G_(1)`, `G_(2)`, `G_(3)` , and `H_(1)`, `H_(2)`, `H_(3)` are the three arithmetic, geometric and harmonic means between two positive numbers `a` and `b(a gt b)`, then which of the following is/are true ?

A

`2G_(1)G_(3)=H_(2)(A_(1)+A_(3))`

B

`A_(2)H_(2)=G_(2)^(2)`

C

`A_(2)G_(2)=H_(2)^(2)`

D

`2G_(1)A_(1)=H_(1)(A_(1)+A_(3))`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to find the three arithmetic means (A1, A2, A3), three geometric means (G1, G2, G3), and three harmonic means (H1, H2, H3) between two positive numbers \( a \) and \( b \) (where \( a > b \)). ### Step 1: Finding the Arithmetic Means The three arithmetic means between \( a \) and \( b \) can be found using the formula for arithmetic progression (AP). If \( A_1, A_2, A_3 \) are the arithmetic means, we can express them as: - \( A_1 = a + d \) - \( A_2 = a + 2d \) - \( A_3 = a + 3d \) Where \( d \) is the common difference. The common difference can be calculated as: \[ d = \frac{a - b}{4} \] Thus, substituting \( d \): - \( A_1 = a + \frac{a - b}{4} = \frac{5a - b}{4} \) - \( A_2 = a + \frac{2(a - b)}{4} = \frac{3a + b}{4} \) - \( A_3 = a + \frac{3(a - b)}{4} = \frac{a + 3b}{4} \) ### Step 2: Finding the Geometric Means The three geometric means between \( a \) and \( b \) can be found using the formula for geometric progression (GP). If \( G_1, G_2, G_3 \) are the geometric means, we can express them as: - \( G_1 = \sqrt{a \cdot b} \) - \( G_2 = \sqrt{G_1 \cdot b} = \sqrt{a \cdot b} \cdot \sqrt{\frac{b}{a}} = b^{3/4} a^{1/4} \) - \( G_3 = \sqrt{G_2 \cdot b} = \sqrt{G_1 \cdot G_2} = \sqrt{b^{3/4} a^{1/4} \cdot b} = a^{1/2} b^{1/2} \) ### Step 3: Finding the Harmonic Means The three harmonic means between \( a \) and \( b \) can be found using the formula for harmonic progression (HP). If \( H_1, H_2, H_3 \) are the harmonic means, we can express them as: \[ H_n = \frac{2ab}{a + b} \] Thus, the harmonic means can be calculated as: - \( H_1 = \frac{2ab}{a + b} \) - \( H_2 = \frac{2ab}{a + b} \) - \( H_3 = \frac{2ab}{a + b} \) ### Conclusion Now, we can summarize the results: - The three arithmetic means are \( A_1, A_2, A_3 \). - The three geometric means are \( G_1, G_2, G_3 \). - The three harmonic means are \( H_1, H_2, H_3 \).
To solve the problem, we need to find the three arithmetic means (A1, A2, A3), three geometric means (G1, G2, G3), and three harmonic means (H1, H2, H3) between two positive numbers \( a \) and \( b \) (where \( a > b \)). ### Step 1: Finding the Arithmetic Means The three arithmetic means between \( a \) and \( b \) can be found using the formula for arithmetic progression (AP). If \( A_1, A_2, A_3 \) are the arithmetic means, we can express them as: - \( A_1 = a + d \) - \( A_2 = a + 2d \) - \( A_3 = a + 3d \) ...
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.1|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE ENGLISH|Exercise ILLUSTRATION 5.2|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE ENGLISH|Exercise ARCHIVES (MATRIX MATCH TYPE )|1 Videos

  • PROBABILITY II

    CENGAGE ENGLISH|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE ENGLISH|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

If g_(1) , g_(2) , g_(3) are three geometric means between two positive numbers a and b , then g_(1) g_(3) is equal to

Let A;G;H be the arithmetic; geometric and harmonic means between three given no. a;b;c then the equation having a;b;c as its root is

If a_(1),a_(2),a_(3) be any positive real numbers, then which of the following statement is not true.

If A_1, A_2, G_1, G_2, ; a n dH_1, H_2 are two arithmetic, geometric and harmonic means respectively, between two quantities aa n db ,t h e na b is equal to A_1H_2 b. A_2H_1 c. G_1G_2 d. none of these

If arithmetic mean of two positive numbers is A, their geometric mean is G and harmonic mean H, then H is equal to

If H_(1) and H_(2) are two harmonic means between two positive numbers a and b (a != b) , A and G are the arithmetic and geometric menas between a and b , then (H_(2)+H_(1))/(H_(2)H_(1)) is

If A_(1), A_(2), A_(3),....A_(51) are arithmetic means inserted between the number a and b, then find the value of ((b + A_(51))/(b - A_(51))) - ((A_(1) + a)/(A_(1) - a))

Let A_1 , G_1, H_1 denote the arithmetic, geometric and harmonic means respectively, of two distinct positive numbers. For n >2, let A_(n-1),G_(n-1) and H_(n-1) has arithmetic, geometric and harmonic means as A_n, G_N, H_N, respectively.

If a_(1),a_(2),a_(3),……a_(87),a_(88),a_(89) are the arithmetic means between 1 and 89 , then sum_(r=1)^(89)log(tan(a_(r ))^(@)) is equal to

Let a_(1), a_(2) ...be positive real numbers in geometric progression. For n, if A_(n), G_(n), H_(n) are respectively the arithmetic mean, geometric mean and harmonic mean of a_(1), a_(2),..., a_(n) . Then, find an expression for the geometric mean of G_(1), G_(2),...,G_(n) in terms of A_(1), A_(2),...,A_(n), H_(1), H_(2),..., H_(n)