# If A, B, C Are in G.P., Prove That: ( a + B + C ) 2 a 2 + B 2 + C 2 = a + B + C a − B + C - Mathematics

If a, b, c are in G.P., prove that:

$\frac{(a + b + c )^2}{a^2 + b^2 + c^2} = \frac{a + b + c}{a - b + c}$

#### Solution

a, b and c are in G.P.

$\therefore b^2 = ac$   .......(1)

$\text { LHS }= \frac{\left( a + b + c \right)^2}{a^2 + b^2 + c^2}$

$= \frac{\left( a + b + c \right)^2}{a^2 - b^2 + c^2 + 2 b^2}$

$= \frac{\left( a + b + c \right)^2}{a^2 - b^2 + c^2 + 2ac} \left[ \text { Using } (1) \right]$

$= \frac{\left( a + b + c \right)^2}{\left( a + b + c \right)\left( a - b + c \right)} \left[ \because \left( a + b + c \right)\left( a - b + c \right) = a^2 - b^2 + c^2 + 2ac \right]$

$= \frac{\left( a + b + c \right)}{\left( a - b + c \right)} =\text { RHS }$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.5 | Q 8.3 | Page 46