JEE Main: GP se AP ka Hidden Rule — Concept Clear in 1 Min 💡

 

❓ Concept

🎬 GP → AP Concept in 59 Sec

Kabhi question bole:
👉 “Numbers GP mein hain, par kuch subtract/add karne ke baad AP ban gaye”

Yeh coincidence nahi hota —
yeh pure logic + standard form ka game hai 🔥

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🖼️ Concept Image

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✍️ Short Explanation

Is type ke questions mein students ki sabse badi galti hoti hai:
❌ numbers ko direct manipulate karna

JEE ka smart approach hai:
👉 GP ko standard form mein likho
👉 AP ki difference condition lagao
👉 a aur r cleanly nikal jaate hain

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🔹 Step 1 — Write GP in Standard Form (CONTROL STEP 🔥)**

Agar

$$x_1,\ x_2,\ x_3,\ x_4$$

GP mein hain, toh hamesha likho:

$$x_1=a,\quad x_2=ar,\quad x_3=ar^2,\quad x_4=ar^3$$

📌 Bas yahin se poora question control mein aa jaata hai.

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🔹 Step 2 — Apply the Given Operation

Question mein jo operation diya ho (usually subtraction/addition), apply karo:

$$(x_1-k_1),\ (x_2-k_2),\ (x_3-k_3),\ (x_4-k_4)$$

Ab:

• Original GP disturb ho chuki hai

• New sequence ko AP banana hai

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🔹 Step 3 — Condition for AP (MOST POWERFUL 💯)**

AP ke liye any one condition enough hoti hai:

Difference form:

$$(x_2-k_2)-(x_1-k_1) = (x_3-k_3)-(x_2-k_2) = (x_4-k_4)-(x_3-k_3)$$

OR middle-term form (fastest 😎):

$$2\times(\text{middle term}) = \text{sum of neighbours}$$

📌 Exam mein middle-term condition sabse clean hoti hai.

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🔹 Step 4 — Convert Condition into Equation

Ab GP terms substitute karo:

$$a,\ ar,\ ar^2,\ ar^3$$

AP condition lagate hi:

• Usually r pehle nikal jaata hai

• Phir a directly mil jaata hai

No guesswork.
No trial-error.

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🔹 Step 5 — Product Shortcut (JEE Favourite 🔥)**

GP ke 4 terms ka product:

$$x_1x_2x_3x_4 = (a)(ar)(ar^2)(ar^3) = a^4r^6$$

📌 Isliye:

• Jab $a$ aur $r$ mil jaaye

• Product direct nikal jaata hai

Often question poochta hai:

$$\frac{1}{\text{constant}}\times (x_1x_2x_3x_4)$$

So final step super fast hota hai.

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✅ Final Takeaway

🧠 GP → AP Golden Rules

• GP ko hamesha standard form mein likho

• Given operation apply karo (−k, +k, etc.)

• AP condition lagao (difference / middle-term)

• Pehle r, phir a nikalo

• Product = $a^4r^6$

Is logic ko follow kar liya,
👉 GP–AP conversion ke JEE questions 1 minute mein finish 🚀

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