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If A,B,C Are in G.P and A,X,B,Y,C Are in A.P Prove that 1/X + 1/Y = 2/B - Mathematics

Sum

If a,b,c are in G.P and a,x,b,y,c are in A.P prove that `1/x + 1/y = 2/b`

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Solution

a, b and c are in G.P.
⇒ b2 = ac
a, x, b, y and c are in A.P.

⇒ 2x = a + b ⇒ x = `(a+b)/2`

2b = x + y ⇒ b = `(x+y)/2`

2y = b + c ⇒ y = `(b+c)/2`
Now,

`1/x+1/y=2/(a+b)+2/(b+c)`

`=(2b+2c+2a+2b)/(ab+ac+b^2+bc)`

`=(2a+2c+4b)/(ab+b^2+b^2+bc)`

`=(2a+2c+4b)/(ab+2b^2+bc)`

`=(2(a+c+2b))/(b(a+2b+c))`

`=2/b`

Concept: Simple Applications - Geometric Progression
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APPEARS IN

Selina Concise Maths Class 10 ICSE
Chapter 11 Geometric Progression
Exercise 11 (C) | Q 8.1 | Page 156
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