# If A, B, C Are in A.P. and A, B, D Are in G.P., Show that A, (A − B), (D − C) Are in G.P. - Mathematics

If a, b, c are in A.P. and a, b, d are in G.P., show that a, (a − b), (d − c) are in G.P.

#### Solution

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

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

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

$\therefore b^2 = ad . . . . . . . (ii)$

$\text { Now }, \left( a - b \right)^2 = a^2 - 2ab + b^2$

$\Rightarrow \left( a - b \right)^2 = a^2 - a\left( a + c \right) + ad \left[ \text { Using } (i)\text { and } (ii) \right]$

$\Rightarrow \left( a - b \right)^2 = a^2 - a^2 - ac + ad$

$\Rightarrow \left( a - b \right)^2 = ad - ac$

$\Rightarrow \left( a - b \right)^2 = a(d - c)$

$\text { Therefore, }a, \left( a - b \right) \text { and } (d - c) \text { are in G . P }.$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.5 | Q 21 | Page 46