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Sequences & Series: Between Between

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

Question
If a,b,ca, b, c are in A.P., p,p′p, p' are the A.M. and G.M. between aa and bb, and q,q′q, q' are the A.M. and G.M. between bb and cc, then
Ap2+q2=p′2+q′2p^2+q^2=p'^2+q'^2
Bpq=p′q′pq=p'q'
Cp2−q2=p′2−q′2p^2-q^2=p'^2-q'^2correct
Dp2+p′2=q2+q′2p^2+p'^2=q^2+q'^2
Solution
Step 1: Here p=a+b2p=\dfrac{a+b}{2}, q=b+c2q=\dfrac{b+c}{2}, p′=abp'=\sqrt{ab}, q′=bcq'=\sqrt{bc}, with 2b=a+c.2b=a+c. Step 2:
p2−q2=(a+b)2−(b+c)24=(a−c)(a+2b+c)4=(a−c)⋅4b4=b(a−c),p^2-q^2=\frac{(a+b)^2-(b+c)^2}{4}=\frac{(a-c)(a+2b+c)}{4}=\frac{(a-c)\cdot 4b}{4}=b(a-c),
 using a+c=2ba+c=2b so a+2b+c=4b.a+2b+c=4b. Step 3:
p′2−q′2=ab−bc=b(a−c).p'^2-q'^2=ab-bc=b(a-c).
 So p2−q2=p′2−q′2.p^2-q^2=p'^2-q'^2. Correct answer: (3)
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