Sequences & Series
Geometric Progression (GP)
Grade None
Question:
<p>If three distinct numbers a,b,c are in G.P. and the equations ax?+2bx+c=0 and dx? + 2ex + f = 0 have acommon root, then which one of the following statements is correct? [JEE (Main) 2019] f (1) d,e, f are in AP. are in GP. ass, c f are in A.P. (4) d,e, f are in G.P. @ 2S, a</p>
d,e,f are in AP.
d,e,f are in GP.
a,s,c are in A.P.
d,e,f are in G.P.
Step-by-Step Solution
Key Concept: Notice the hidden common ratio first. Once the ratio is pinned down, the rest of the progression falls into place very quickly.
Notice that the real unlock is the common ratio. A clever move here is to write consecutive terms in powers of the ratio and then use the given condition to collapse the algebra. Now, we invoke the power of the G.P. identities for the nth term and the sum, simplify patiently, and verify that the final answer is $1$.
Correct Answer: 1