From given A.P i.e., a $b^{2}$ $c^{3}$ , $a^{2}$ $b^{3}$ $c^{4}$ , $a^{3}$ $b^{4}$ $c^{5}$ , get the value of abc __

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Solution

Observe that a + b + c is a part of AM of a, b, c i.e., $\frac{(a + b + c)}{3}$ as AM $\geq$ GM & HM,

We can get minimum value of AM from this relation : AM $\geq$ GM

Using AM $\geq$ GM $\Rightarrow$ a + b + c $\geq$ 3 ${(a b c)}^{\frac{1}{3}}$ ------------(1)

We need value of abc

Given a $b^{2}$ $c^{3}$ , $a^{2}$ $b^{3}$ $c^{4}$ , $a^{3}$ $b^{4}$ $c^{5}$ are in A.P.

$\frac{2 a^{2} b^{3} c^{4}}{a b^{2} c^{3}}$ = $\frac{a b^{2} c^{3}}{a b^{2} c^{3}}$ + $\frac{a^{3} b^{4} c^{5}}{a b^{2} c^{3}}$

$\Rightarrow$ 2abc = 1 + $a^{2}$ $b^{2}$ $c^{2}$

$\Rightarrow$ ${(a b c)}^{2}$ - 2abc + 1 = 0

$\Rightarrow$ ${(a b c - 1)}^{2}$ = 0

So, abc = 1

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Similar questions

Q. STATEMENT-1 : If

a b^{2} c^{3}, a^{2} b^{3} c^{4}, a^{3} b^{4} c^{5}

are in A.P.

(a, b, c > 0),

then the minimum value of a + b + c is 3.

STATEMENT-2 : Arithmetic mean of any two numbers is greater than geometric mean of the numbers.

If a $b^{2}$ $c^{3}$ , $a^{2}$ $b^{3}$ $c^{4}$ , $a^{3}$ $b^{4}$ $c^{5}$ are in A.P. (a, b, c > 0) then the minimum value of a + b + c is __

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A coin is tossed $150$ times and the outcomes are recorded. The frequency distribution of the outcomes $H$ (i.e, head) and $T$ (i.e, tail) is given below :

Outcome	$H$	$T$
Frequency	$85$	$65$

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Q. A coin is tossed

400

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From given A.P i.e., a b2 c3, a2 b3 c4, a3 b4 c5, get the value of abc __

From given A.P i.e., a $b^{2}$ $c^{3}$ , $a^{2}$ $b^{3}$ $c^{4}$ , $a^{3}$ $b^{4}$ $c^{5}$ , get the value of abc __