Does there exist a geometric progression containin

Question

Does there exist a geometric progression containing 27, 8 and 12 as three of its terms ? If it exits, how many such progressions are possible ?

Aditi Chauhan · Accepted Answer

Hello Student,

Please find the answer to your question

If possible let for a G. P.

T_p= 27 = AR^p-1…………. (1)

T_q = 8 = AR^q-1 ………… (2)

T_r = 12 = AR^r-1 ………… (3)

From (1) and (2)

R ^p-q= 27/8 ⇒ R^p– ^q = (3/2)³ ……….. (4)

From (2) and (3) ;

R^q– ^r = 8/12 ⇒ R^{q – r} = (3/2) -1 ………….. (5)

From (4) and (5)

R = 3/2 ; p-q = 3; q – r = -1

p – 2q + r = 4; p, q, r ∈ N ………… (6)

As there can be infinite natural numbers for p, q, and r to satisfy equation (6)

∴ There can be infinite G. P’ s.

Thanks
Aditi Chauhan
askIITians Faculty

Does there exist a geometric progression containing 27, 8 and 12 as three of its terms ? If it exits, how many such progressions are possible ?