Find three numbers a, b, c, between 2 and 18 such that (i) their sum is 25 (ii) the numbers 2, a, b

Q: Wait, why do we reject the negative root or the value $a = 17$ here?

Great question! In JEE problems, constraints are just as important as the equations themselves. When we found $a = 17$, we calculated $b = 2(17) - 2 = 32$. The problem explicitly states that all numbers must be less than $18$. Since $32 > 18$, this solution violates the boundary condition. Always check your final values against the initial constraints!

Q: Can I use the property $b^2 = ac$ for the A.P. part?

No, that is a common trap! The property $b^2 = ac$ applies to Geometric Progressions. For an Arithmetic Progression $2, a, b$, the correct property is $2a = 2 + b$ (or $a - 2 = b - a$). Mixing up A.P. and G.P. properties is a classic mistake, so keep your definitions distinct.

Q: Is there a faster way to solve this without substitution?

While you could try to solve it using properties of means, substitution is the most reliable method for this level of complexity. It reduces the system to a single-variable quadratic equation, which is the standard, robust way to ensure you don't miss any solutions.

Question

Find three numbers $a, b, c$, between $2$ and $18$ such that (i) their sum is $25$ (ii) the numbers $2, a, b$ sare consecutive terms of an A.P. and (iii) the numbers $b, c, 18$ are consecutive terms of a G.P.

Sigma Percentile · Accepted Answer

### The Geometry of Constraints: A Mathematical Journey

Imagine you are standing in a vast, open field of numbers. You are looking for three specific values, $a$, $b$, and $c$.

These numbers are trapped within a strict boundary: $2 < a, b, c < 18$. This constraint acts as a silent guardian that will help us filter out invalid solutions later on.

### Phase 1: Translating the Clues

We are given three powerful pieces of information. First, the sum of our numbers is $a + b + c = 25$.

Second, the sequence $2, a, b$ forms an Arithmetic Progression (A.P.). Third, the sequence $b, c, 18$ forms a Geometric Progression (G.P.).

For the A.P., the common difference must be constant, implying $a - 2 = b - a$. Rearranging this gives us:

$$b = 2a - 2$$

For the G.P., the common ratio must be constant, implying $\frac{c}{b} = \frac{18}{c}$. By cross-multiplying, we arrive at the relation:

$$c^2 = 18b$$

### Phase 2: The Algebraic Dance

We can reduce our system to a single variable. Substituting our expression for $b$ into the sum $a + b + c = 25$, we get:

$$a + (2a - 2) + c = 25$$

Simplifying this yields $3a - 2 + c = 25$, which allows us to express $c$ in terms of $a$:

$$c = 27 - 3a$$

Now, we substitute these expressions for $b$ and $c$ into our G.P. equation, $c^2 = 18b$:

$$(27 - 3a)^2 = 18(2a - 2)$$

### Phase 3: The Quadratic Showdown

We simplify the equation by factoring out a $3$ from the left side, which becomes $3^2 = 9$ when squared:

$$9(9 - a)^2 = 36(a - 1)$$

Dividing both sides by $9$, we obtain:

$$(9 - a)^2 = 4(a - 1)$$

Expanding both sides results in $81 - 18a + a^2 = 4a - 4$. Rearranging into standard quadratic form, we get:

$$a^2 - 22a + 85 = 0$$

Factoring the quadratic, we find $(a - 17)(a - 5) = 0$. This yields two potential paths: $a = 17$ or $a = 5$.

### Phase 4: The Final Inspection

We must now return to our initial boundary condition: $2 < a, b, c < 18$.

If $a = 17$, then $b = 2(17) - 2 = 32$. Since $32 > 18$, this path violates our constraint and must be rejected.

If $a = 5$, then $b = 2(5) - 2 = 8$, and $c = 27 - 3(5) = 12$.

Checking these values, we see $5 + 8 + 12 = 25$. The A.P. $2, 5, 8$ has a common difference of $3$, and the G.P. $8, 12, 18$ has a common ratio of $1.5$.

The values satisfy all conditions. The final set is **$(a, b, c) = (5, 8, 12)$**.

Animated Solution for Mathematics - Sequence and Series: Find three numbers $a, b, c$ , between $2$ and $18$ such that (i) their sum is $25$ (ii) the numbers $2, a, b$ sare consecutive terms of an A.P. and (iii) the numbers $b, c, 18$ are consecutive terms of a G.P.

Visualized Solution

Understanding the Problem Constraints

Applying the A.P. Property

Applying the G.P. Property

Expressing $c$ in terms of $a$

Forming the Quadratic Equation

Simplifying the Equation (Part 1)

Simplifying the Equation (Part 2)

Solving for $a$

Verifying $a = 17$

Verifying $a = 5$ and Final Answer

The Sigma Insight: Geometric Progression (G.P.)

The Geometry of Constraints

Phase 1

Phase 2

Phase 3

Phase 4

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