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Question

If a+bxa-bx=b+cxb-cx=c+dxc-dx (x ≠ 0), then show that a, b, c and d are in G.P.

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Solution

Given:a+bxa-bx=b+cxb-cx=c+dxc-dxNow, a+bxa-bx=b+cxb-cxApplying componendo and dividendoa+bx+a-bxa+bx-a-bx=b+cx+b-cxb+cx-b-cx2a2bx = 2b2cxab=bcSimiliarly, b+cx+b-cxb+cx-b-cx=c+dx+c-dxc+dx-c-dx bc = cdTherefore, a, b, c and d are in G.P.

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