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Maths Question 6 – JEE-MAIN 2025

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Let x1,x2,x3,x4 be in a geometric progression. If 2,7,9,5 are subtracted respectively from x1,x2,x3,x4, then the resulting numbers are in an arithmetic progression. Then the value of 124(x1x2x3x4) is:

Represent the geometric progression (GP) terms using a first term a and common ratio r, and then form the arithmetic progression (AP) terms by subtracting the given constants.

Step 1: Set up GP and AP terms✦ Active

Let the geometric progression be x1=a,x2=ar,x3=ar2,x4=ar3. The terms after subtraction are x1′=a−2, x2′=ar−7, x3′=ar2−9, x4′=ar3−5. These terms are in an arithmetic progression.

Step 2: Apply AP conditions and solve for a and r○ Expand

Using the AP property x2′−x1′=x3′−x2′:

(ar−7)−(a−2)=(ar2−9)−(ar−7) a(r−1)−5=ar(r−1)−2 −a(r−1)2=3(1)

Using the AP property x3′−x2′=x4′−x3′:

(ar2−9)−(ar−7)=(ar3−5)−(ar2−9) ar2−ar−2=ar3−ar2+4 −ar(r−1)2=6(2)

Dividing equation (2) by (1) (assuming r≠1):

−ar(r−1)2−a(r−1)2=63⟹r=2

Substitute r=2 into equation (1):

−a(2−1)2=3⟹−a=3⟹a=−3

Thus, a=−3 and r=2. The GP terms are x1=−3,x2=−6,x3=−12,x4=−24.

💡 Teacher's Secret Hint

Remember to check the case r=1 to ensure the division is valid. In this case, r=1 leads to a contradiction, so it's safe to divide.

Step 3: Calculate the required value○ Expand

The product of the GP terms is x1x2x3x4=a⋅ar⋅ar2⋅ar3=a4r6.

x1x2x3x4=(−3)4(2)6=81×64=5184

The value to be found is 124(x1x2x3x4):

124(5184)=216
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