11TH•Mathematics•Sequences And Series•6.3 Arithmetic Mean

6.3 Arithmetic Mean (A.M.)

Arithmetic mean between two numbers and inserting n A.M.s.

$A$ is the A.M. between $a$ and $b$ if $a, A, b$ are in A.P., so

A = \dfrac{a + b}{2}

If $A_1, \ldots, A_n$ are $n$ A.M.s between $a$ and $b$ , then $a, A_1, \ldots, A_n, b$ form an A.P.

Find three A.M.s between $\sqrt{2}$ and $3\sqrt{2}$ .

Solution

$\sqrt{2}, A_1, A_2, A_3, 3\sqrt{2}$ is an A.P. with $n = 5$ :

3\sqrt{2} = \sqrt{2} + 4d \implies d = \dfrac{1}{\sqrt{2}}

A_1 = \dfrac{3}{\sqrt{2}},\quad A_2 = 2\sqrt{2},\quad A_3 = \dfrac{5}{\sqrt{2}}

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