Let $x, y, z$ be three positive real numbers in a geometric progression such that $x<y<z$. If $5 x, 16 y$, and $12 z$ are in an arithmetic progression then the common ratio of the geometric progression is

We have three positive real numbers $x, y, z$ that form a geometric progression (GP) where $x < y < z$.

In a geometric progression, each term is obtained by multiplying the previous term by a constant called the common ratio (r). So if $x, y, z$ are in GP, then:

$y = xr$ and $z = yr = xr^2$

This also means $y^2 = xz$ (this is a key property we'll use!)

We're also told that $5x, 16y, 12z$ are in arithmetic progression (AP).

In an arithmetic progression, the middle term is the average of the first and third terms.

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Since $5x, 16y, 12z$ are in AP, the middle term $16y$ must be the average of the first and third terms:

$16y = \dfrac{5x + 12z}{2}$

$32y = 5x + 12z$ ... (1)

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Since $x, y, z$ are in GP:

$y^2 = xz$ ... (2)

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From equation (1): $32y = 5x + 12z$

$(32y)^2 = (5x + 12z)^2$

$1024y^2 = 25x^2 + 120xz + 144z^2$

Since $y^2 = xz$ from equation (2), we can substitute:

$1024(xz) = 25x^2 + 120xz + 144z^2$

$1024xz = 25x^2 + 120xz + 144z^2$

$1024xz - 120xz = 25x^2 + 144z^2$

$904xz = 25x^2 + 144z^2$

$25x^2 + 144z^2 - 904xz = 0$

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$25x^2 - 904xz + 144z^2 = 0$

We need to split the middle term. We look for two numbers that multiply to $(25)(144) = 3600$ and add to $-904$.

We find: $-900$ and $-4$ work because:

$(-900) + (-4) = -904$ and $(-900) \times (-4) = 3600$

$25x^2 - 900xz - 4xz + 144z^2 = 0$

$25x(x - 36z) - 4z(x - 36z) = 0$

$(25x - 4z)(x - 36z) = 0$

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This gives us two possibilities:

$25x - 4z = 0$ which gives $x = \dfrac{4z}{25}$

$x - 36z = 0$ which gives $x = 36z$

Since we're told $x < y < z$ and they're all positive, we need $x < z$.

Option 1: $x = \dfrac{4z}{25}$ means $x < z$

Option 2: $x = 36z$ means $x > z$

Therefore: $x = \dfrac{4z}{25}$

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In a GP, if the common ratio is $r$, then:

$y = xr$

$z = xr^2$

From $z = xr^2$ and $x = \dfrac{4z}{25}$:

$z = \dfrac{4z}{25} \cdot r^2$

$1 = \dfrac{4r^2}{25}$

$25 = 4r^2$

$r^2 = \dfrac{25}{4}$

$r = \dfrac{5}{2}$ (taking the positive root since $x < y < z$)

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Therefore, the common ratio is $\dfrac{5}{2}$.