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Show p, q, r, s in GP from Ratios with 5^x

JEE Maths question with a full step-by-step solution.

Question
If p+q⋅5xp−q⋅5x=q+r⋅5xq−r⋅5x=r+s⋅5xr−s⋅5x\frac{p+q\cdot 5^x}{p-q\cdot 5^x}=\frac{q+r\cdot 5^x}{q-r\cdot 5^x}=\frac{r+s\cdot 5^x}{r-s\cdot 5^x}, then p,q,r,sp, q, r, s are in
AA.P.
BG.P.correct
CH.P.
Dnone of these
Solution
Step 1: Apply componendo and dividendo to the first equality:
(p+q5x)+(p−q5x)(p+q5x)−(p−q5x)=(q+r5x)+(q−r5x)(q+r5x)−(q−r5x) ⇒ 2p2q⋅5x=2q2r⋅5x ⇒ pq=qr.\frac{(p+q5^x)+(p-q5^x)}{(p+q5^x)-(p-q5^x)}=\frac{(q+r5^x)+(q-r5^x)}{(q+r5^x)-(q-r5^x)}\ \Rightarrow\ \frac{2p}{2q\cdot 5^x}=\frac{2q}{2r\cdot 5^x}\ \Rightarrow\ \frac{p}{q}=\frac{q}{r}.
 Step 2: The same on the second equality gives qr=rs.\dfrac{q}{r}=\dfrac{r}{s}. Step 3: So pq=qr=rs ⇒ p,q,r,s\dfrac{p}{q}=\dfrac{q}{r}=\dfrac{r}{s}\ \Rightarrow\ p, q, r, s are in G.P. Correct answer: (2)
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