$a$ $a$ i s The arithmetic mean of two positive numbers $b and c$ $b and c$ . $G_{1}$ $G_1$ and $G_{2}$ $G_2$ are the geometric mean between the same positive numbers $b and c$ $b and c$ so prove that $G_{1}^{3} + G_{2}^{3}$ $G_1^3+G_2^3$ = $2 a b c$ $2abc$ ?

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$a$ $a$ i s The arithmetic mean of two positive numbers $b and c$ $b and c$ . $G_{1}$ $G_1$ and $G_{2}$ $G_2$ are the geometric mean between the same positive numbers $b and c$ $b and c$ so prove that $G_{1}^{3} + G_{2}^{3}$ $G_1^3+G_2^3$ = $2 a b c$ $2abc$ ?

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Answer 1 · 2018-08-09T11:50:59

Please see below.

Explanation:

Here,

$a is the AM of b and c \Rightarrow \frac{b + c}{2} = a \Rightarrow b + c = 2 a \to (1)$ $color(blue)("a is the AM of b and c " =>(b+c)/2=a=>b+c=2ato(1)$

Now, $G_{1} and G_{2} are the GM between b and c.$ $color(red)(G_1 and G_2 " are the GM between b and c."$

So, $b, G_{1}, G_{2}, c are in GP$ $b,G_1,G_2,c " are in GP"$

Let $r be the common ratio and b is the first term.$ $r " be the common ratio and b is the first term."$

So, $b, b r, b r^{2}, b r^{3} are in GP$ $b, br, br^2,br^3 " are in GP"$

$i.e. color(blue)(G_1=br ,G_2=br^2 ,c=br^3......to(2)$

Let us take LHS.

$LHS=(G_1)^3+(G_2)^3$

$LHS=(br)^3+(br^2)^3$

$LHS=b^3r^3+b^3r^6$

$LHS=b^2color(blue)((br^3))+b(color(blue)(br^3))^2$

$LHS=b^2color(blue)((c))+bcolor(blue)((c)^2....to["using " (2) ]$

$LHS=bc{color(blue)(b+c)}$

$LHS=bc{color(blue)(2a)}...............to["using " (1)]$

$LHS=2abc$

$LHS=RHS$

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$a$ $a$ i s The arithmetic mean of two positive numbers $b and c$ $b and c$ . $G_{1}$ $G_1$ and $G_{2}$ $G_2$ are the geometric mean between the same positive numbers $b and c$ $b and c$ so prove that $G_{1}^{3} + G_{2}^{3}$ $G_1^3+G_2^3$ = $2 a b c$ $2abc$ ?

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aaa i s The arithmetic mean of two positive numbers b and cbandcb and c . G_1G1G_1 and G_2G2G_2 are the geometric mean between the same positive numbers b and cbandcb and c so prove that G_1^3+G_2^3G31+G32G_1^3+G_2^3=2abc2abc2abc ?

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$a$ $a$ i s The arithmetic mean of two positive numbers $b and c$ $b and c$ . $G_{1}$ $G_1$ and $G_{2}$ $G_2$ are the geometric mean between the same positive numbers $b and c$ $b and c$ so prove that $G_{1}^{3} + G_{2}^{3}$ $G_1^3+G_2^3$ = $2 a b c$ $2abc$ ?