Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

# If A, B, C Are in G.P., Prove that the Following is Also in G.P.: A2 + B2, Ab + Bc, B2 + C2 - Mathematics

If a, b, c are in G.P., prove that the following is also in G.P.:

a2 + b2, ab + bc, b2 + c2

#### Solution

a, b and c are in G.P.
∴ $b^2 = ac . . . . . . . (1)$

$\left( ab + bc \right)^2 = \left( ab \right)^2 + 2a b^2 c + \left( bc \right)^2$

$\Rightarrow \left( ab + bc \right)^2 = \left( ab \right)^2 + a b^2 c + a b^2 c + \left( bc \right)^2$

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

$\Rightarrow \left( ab + bc \right)^2 = a^2 \left( b^2 + c^2 \right) + b^2 \left( b^2 + c^2 \right)$

$\Rightarrow \left( ab + bc \right)^2 = \left( b^2 + c^2 \right)\left( a^2 + b^2 \right)$

$\text { Therefore }, \left( a^2 + b^2 \right), \left( b^2 + c^2 \right) \text { and }\left( ab + bc \right) \text { are also in G . P } .$

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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 20 Geometric Progression
Exercise 20.5 | Q 10.3 | Page 46