if Sn denoted sum to n terms of G.P. prove that (S10 - S20)2 = S10(S30 - S20)

As Sn is sum of n terms which is given by the formula
Sn=a(1-rn)/(1-r)
Where a is first term of GP and r is common ratio
We have to prove, (S10 - S20)2 = S10(S30 - S20)
Now LHS
(S10 - S20)2
=(a(1-r10)(1-r)-a(1-r20)(1-r))2=(a(r20-r10)(1-r))2=(ar10(1-r10)(1-r))2=a2r20(1-r10)2(1-r)2
Now RHS
=  S10(S30 - S20)
=a(1-r10)(1-r)(a(1-r30)(1-r)- a(1-r20)(1-r))=a(1-r10)(1-r)( a(r20-r30)(1-r))=a2r20(1-r10)2(1-r)2
LHS=RHS
Hence proved

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