GP Product and Sum Relation Trick

 

❓ Concept Question

In a G.P., if the product of first three terms is fixed, how can we find the range of the sum of first three terms?

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🖼 Concept Image

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✍️ Short Concept

This concept is based on:

👉 Geometric Progression (G.P.)
👉 Middle term property
👉 AM-GM inequality
👉 Range of expressions.

Main idea:

For three terms in G.P.:

$$\boxed{ \frac ar,\ a,\ ar }$$

their product becomes:

$$\boxed{ \left(\frac ar\right)(a)(ar)=a^3 }$$

So ratio cancels automatically.

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🔷 Step 1 — Standard Form of 3 G.P. Terms 💯

Three terms of G.P. are generally written as:

$$\boxed{ \frac ar,\ a,\ ar }$$

Why this form?

✔ Middle term stays clean

✔ Ratio handling becomes easy

✔ Product simplifies directly

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🔷 Step 2 — Product Trick

For these three terms:

$$\left(\frac ar\right)\cdot a \cdot(ar)$$
$$=a^3$$

Important observation:

$$\boxed{ \text{Common ratio } r \text{ cancels out} }$$

Thus:

✔ Fixed product ⇒ fixed middle term.

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🔷 Step 3 — Sum of Three Terms

Sum becomes:

$$S=\frac ar+a+ar$$

Take $a$ common:

$$S=a\left(r+1+\frac1r\right)$$

So the entire range depends on:

$$\boxed{ r+\frac1r }$$

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🔷 Step 4 — Golden AM-GM Trick ✨

For positive $r$:

$$\boxed{ r+\frac1r\ge2 }$$

Equality occurs at:

$$r=1$$

Therefore:

$$S_{\min}=a(2+1)=3a$$

This gives the minimum possible sum.

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🔷 Step 5 — JEE Observation 🚨

Whenever question says:

✔ Product of G.P. terms fixed

Immediately think:

$$\boxed{ \text{Middle term fixed} }$$

Then reduce sum into:

$$r+\frac1r$$

and apply:

$$\boxed{ x+\frac1x\ge2 }$$

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✅ Final Takeaway

For three G.P. terms:

$$\boxed{ \frac ar,\ a,\ ar }$$

Important formulas:

Product

$$\boxed{ =a^3 }$$

Sum

$$\boxed{ =a\left(r+1+\frac1r\right) }$$

Key inequality

$$\boxed{ r+\frac1r\ge2 }$$

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🌟 Golden JEE Insight

Most G.P. range questions reduce to:

$$\boxed{ x+\frac1x }$$

Once you spot this pattern:

✔ Minimum value questions become very fast

✔ No need for lengthy calculus

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📚 Related Topics

• Find Polynomial Value Using Root Formation Method
• Find Sum of Fourth Powers Using Symmetric Identities
• Shortest Distance Between Lines and Ellipse Relation