Assertion: The value of x+y+z is 15 if a, x, y, z b are in...

Assertion: The value of $x+y+z$ is 15 if $a, x, y, z$ $b$ are in A.P., while the value of $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}$ is $\frac{5}{3}$ if $a, x, y, z, b$ are in H.P. The values of $a$ and $b$ are 9,1 respectively. Reason: The sum of $n$ A.M.s between two quantities is equal to $n$ times their single mean.

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Key Concepts

Harmonic Progression (HP)

A harmonic progression is a sequence of numbers such that their reciprocals form an arithmetic progression. This transformation allows problems involving harmonic means and sum of reciprocals to be analyzed by first considering the properties of arithmetic progressions.

Arithmetic Mean (A.M.) and Its Properties

The arithmetic mean of two numbers is the simple average, and inserting arithmetic means between two given numbers forms an arithmetic progression. A key property in these contexts is that the sum of multiple arithmetic means between two numbers equals the number of means multiplied by the single arithmetic mean. This property is often used to simplify and solve problems related to sequences and series.

Arithmetic Progression (AP)

An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. This constant, known as the common difference, allows for the use of summation formulas and the systematic calculation of intermediate values (arithmetic means) within the sequence.

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