Question

if
$\frac{2}{b} = \frac{1}{a} + \frac{1}{c}$
then a,b,c are in​

Answer 1

$\underline\blue{\bold{Given \: Question :- }}$

$\bf \:if \: \dfrac{2}{b} = \dfrac{1}{a} + \dfrac{1}{c}$

then a, b, c are in

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$\begin{gathered}\Large{\bold{\pink{\underline{Basic \: Concepts \::}}}} \end{gathered}$

• An arithmetic progression (AP) is a sequence of numbers in which each succeeding number is obtained either by adding or subtracting a specific number called common difference. The general form of AP is: a, a + d, a + 2d, …

• A geometric progression (GP) is a sequence of numbers in which each succeeding number is obtained multiplying a specific number called common ratio. The general form of GP is: a, ar, ar^2, ,….

• A sequence of numbers is said to be a harmonic progression if the reciprocal of those numbers are in AP. In simple terms, a,b,c,d,e,f are in HP if 1/a, 1/b, 1/c, 1/d, 1/e, 1/f are in AP.

• For two terms ‘a’ and ‘b’,
• Harmonic Mean = (2 a b) / (a + b)
• For two numbers, if A, G and H are respectively the arithmetic, geometric and harmonic means, then
• A ≥ G ≥ H
• A H = G^2, i.e., A, G, H are in GP

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$\begin{gathered}\Large{\bold{\purple{\underline{CaLcUlAtIoN\::}}}} \\ \end{gathered}$

$\bf \:\dfrac{2}{b} = \dfrac{1}{a} + \dfrac{1}{c}$

$\bf\implies \:\dfrac{1}{b} + \dfrac{1}{b} = \dfrac{1}{a} + \dfrac{1}{c}$

$\bf\implies \:\dfrac{1}{b} - \dfrac{1}{a} = \dfrac{1}{c} - \dfrac{1}{b}$

$\bf\implies \:\dfrac{1}{a}, \dfrac{1}{b}, \dfrac{1}{c} \: are \: in \: AP$

$\bf\implies \:a,b,c \: are \: in \: HP.$

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