# If A, B, C Are in A.P. and A, X, B and B, Y, C Are in G.P., Show that X2, B2, Y2 Are in A.P. - Mathematics

If a, b, c are in A.P. and a, x, b and b, y, c are in G.P., show that x2, b2, y2 are in A.P.

#### Solution

$\text { a, b and c are in A . P } .$

$\therefore 2b = a + c . . . . . . . (i)$

$\text { a, x and b are in G . P } .$

$\therefore x^2 = ab . . . . . . . (ii)$

$\text { And, b, y and c are also in G . P } .$

$\therefore y^2 = bc . . . . . . . (iii)$

$\text { Now, putting the values of a and c: }$

$\Rightarrow 2b = \frac{x^2}{b} + \frac{y^2}{b}$

$\Rightarrow 2 b^2 = x^2 + y^2$

$\text { Therefore,} x^2 , b^2 \text { and } y^2 \text { are also in A . P } .$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.5 | Q 20 | Page 46