Verified Solution • 2025–26 Session moderate

QJEEM21Mabc10: JEE Main 2021 Mathematics - Class 11 JEEM21Mabc10 Solution: Three numbers are in an increasing geometric progression with…

Three numbers are in an increasing geometric progression with common ratio r. If the middle number is doubled, then the new numbers are in an arithmetic progression with common difference d. If the fourth term of GP is 3 r2, then r2 −- d is equal to :

Multiple Choice Options

  • A 7 −- 7 3\sqrt 3
  • B 7 + 3\sqrt 3
  • C 7 −- 3\sqrt 3
  • D 7 + 3 3\sqrt 3

Step-by-Step Verified Solution

Let numbers be ar{a \over r} , a, ar →\to G.P.

ar{a \over r} , 2a, ar →\to A.P. ⇒\Rightarrow 4a = ar{a \over r} + ar ⇒\Rightarrow r + 1r{1 \over r} = 4

r = 2 ±\pm 3\sqrt 3

4th form of G.P. = 3r2 ⇒\Rightarrow ar2 = 3r2 ⇒\Rightarrow a = 3

r = 2 + 3\sqrt 3 , a = 3, d = 2a −- ar{a \over r} = 3 3\sqrt 3

r2 −- d = (2 + 3\sqrt 3 )2 −- 3 3\sqrt 3

= 7 + 4 3\sqrt 3 −- 3 3\sqrt 3

= 7 + 3\sqrt 3

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