If x, 2y, 3z are in A.P., where the distinct numbers x, y, z are in G.P. then the common ratio of the G.P. is

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x, 2y, 3z

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Concept understanding — Geometric Progression Arithmetic Progression

Comparing Arithmetic and Geometric Progressions

When you look at a sequence of numbers, the very first question to ask is: how does one term turn into the next? Two patterns cover almost every sequence you'll meet in Class 11 — adding a fixed number each time, or multiplying by a fixed number each time.

Two different rules of motion

An Arithmetic Progression (AP) moves by addition: each term is the previous term plus a constant common difference $d$ . If the first term is $a$ , the sequence is $a,\ a+d,\ a+2d,\ \dots$ , so the growth is a straight line — linear.
A Geometric Progression (GP) moves by multiplication: each term is the previous term times a constant common ratio $r$ . If the first term is $a$ , the sequence is $a,\ ar,\ ar^2,\ \dots$ , so the growth curves sharply upward (or shrinks toward zero) — exponential, not linear.

Note

This page is not the place to re-derive the individual formulas from scratch — the full nth-term and sum derivations for AP live under the Arithmetic Progression concept, and the equivalent derivations for GP live under the Geometric Progression concept. This page exists purely to help you tell the two patterns apart and choose the right one quickly.

The fastest way to tell them apart

Take any three consecutive terms $t_1, t_2, t_3$ of the sequence you're given:

Test for AP: compute $t_2-t_1$ and $t_3-t_2$ . If they're equal, you have an AP, and that equal value is $d$ .
Test for GP: compute $\dfrac{t_2}{t_1}$ and $\dfrac{t_3}{t_2}$ . If they're equal, you have a GP, and that equal value is $r$ .

If neither test succeeds, the sequence is neither an AP nor a GP on its own — it may still be built from a combination of the two (an arithmetico-geometric sequence), which needs its own separate technique.

Side-by-side

Feature	Arithmetic Progression	Geometric Progression
Rule to get the next term	Add $d$	Multiply by $r$
nth term	$a+(n-1)d$	$a\,r^{\,n-1}$
Sum of $n$ terms	$\dfrac{n}{2}[2a+(n-1)d]$	$a\,\dfrac{r^n-1}{r-1}$ (for $r\neq1$ )
Shape of growth	Straight line	Curve (exponential)
Allowed values of $d$ / $r$	Any real number, including $0$	Any real number except $0$

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The problem gives two conditions: $x, 2y, 3z$ are in arithmetic progression, and $x, y, z$ (distinct) are in geometric progression. Using the AP property $2 \cdot (2y) = x + 3z$ and the GP property $y^2 = xz$ , we solve for the common ratio $r = y/x$ . The valid distinct ratio is $\boxed{\frac{1}{3}}$ , which corresponds to option (B).

We have two sequences here, and the trick is to connect them through the common terms $x, y, z$ . Let’s start with what each progression tells us.

Geometric Progression (GP): If $x, y, z$ are in GP, then the ratio between consecutive terms is constant. Let that common ratio be $r$ . So:

y = xr, \quad z = xr^2

This is the cleanest way to express all three in terms of $x$ and $r$ . Since $x, y, z$ are distinct, $r \neq 1$ and $x \neq 0$ (otherwise all would be zero, not distinct).

Arithmetic Progression (AP): The terms $x, 2y, 3z$ are in AP. For three numbers $a, b, c$ to be in AP, the middle term is the average of the other two:

2b = a + c

Here $a = x$ , $b = 2y$ , $c = 3z$ . So:

2 \cdot (2y) = x + 3z \quad \Rightarrow \quad 4y = x + 3z

Now substitute the GP expressions into this AP equation.

Substitute $y = xr$ and $z = xr^2$ into $4y = x + 3z$ :

4(xr) = x + 3(xr^2)

Since $x \neq 0$ , divide through by $x$ :

4r = 1 + 3r^2

Rearrange into a quadratic equation:

3r^2 - 4r + 1 = 0

Solve the quadratic:

r = \frac{4 \pm \sqrt{16 - 12}}{6} = \frac{4 \pm 2}{6}

So:

r = \frac{4 + 2}{6} = 1 \quad \text{or} \quad r = \frac{4 - 2}{6} = \frac{1}{3} …

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Q.If x,2y,3zx, 2y, 3zx,2y,3z are in A.P., where the distinct numbers x,y,zx, y, zx,y,z are in G.P. then the common ratio of the G.P. is (A) 333 (B) 13\dfrac{1}{3}31​ (C) 222 (D) 12\dfrac{1}{2}21​

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Q.If $x, 2y, 3z$ are in A.P., where the distinct numbers $x, y, z$ are in G.P. then the common ratio of the G.P. is
(A) $3$
(B) $\dfrac{1}{3}$
(C) $2$
(D) $\dfrac{1}{2}$

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