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It a(1) , a(2) , a(3) a(4) be in G.P. t...

It ` a_(1) , a_(2) , a_(3) a_(4)` be in G.P. then prove that `(a_(2)-a_(3))^(2) + (a_(3) - a_(1))^(2) + (a_(4) +a_(2))^(2) = (a_(1)-a_(4))^(2)`

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While proving conditional identitonal when given numbers are in A.P., G.P. or H.P., we express.,
the terms of say , G.P. as a function of first term and common ratio
Let r be common ratio . Then ` a_(2) = a_(1) r. a_(3) = a_(1) r^(2) , a_(4) = a_(1)r^(3)`
LHS = `(a_(2) - a_(3))^(2) + (a_(2) - a_(1))^(2) + (a_(4) -a_(2))^(2)`
` = (a_(1) r - a_(1) r^(2))^(2) + (a_(1) r^(2) - a_(1))^(2) + (a_(1) r^(3) -a_(1) r)^(2)`
` a_(1)^(2) {(r-r^(2))^(2) + (r^(2) -1)^(2) + (r^(3) -r)^(2)}`
` a_(1)^(2) { r ^(2) - 2r^(3) + r^(4) + r^(4) - 2r^(2) + 1 + r^(6) - 2r^(4) + r^(2)}`
` a_(1)^(2) {r^(6) - 2r^(3) + 1} = a_(1)^(2)(r^(3) -1)^(2) = (a_(1)r^(3) -a_(1))^(2)= (a_(4) -a_(1))^(2)`
RHS
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