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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

\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, s

are in

AA.P.

BG.P.correct

CH.P.

Dnone of these

Solution

Step 1: Apply componendo and dividendo to the first equality:

\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

\dfrac{q}{r}=\dfrac{r}{s}.

Step 3: So

\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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