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As we are given a, b,c are in G.P lets assume, a=br; b=b; c=br
Where r→common ratio between the terms such that ratio of cb=ba
Now, we calculate value of a2 + b2, ab+ bc, b2 + c2 to prove that these are in G.P
1st term=a2 + b2=(br)2+b2=b2(1+r2)r2
2nd term ab+bc=br×b+b×br=b2(1+r2)r
3rd term b2 + c2=b2+b2r2=b2(1+r2)
3rdterm2ndterm=b2(1+r2)b2(1+r2)r=b2(1+r2)×rb2(1+r2)=r …(1)
2ndterm1stterm=b2(1+r2)r×r2b2(1+r2)=r …(2)
From eq (1) & (2)common ratios are same so a2 + b2, ab+ bc, b2 + c2 in G.P
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