A, B, C are in G P and A, X, B, Y, C are A P prove 1X+1Y=2B - Maths - Sequences and Series

Question

A, B, C are in G P and A, X, B, Y, C are A P prove 1 X + 1 Y = 2 B

Abhishek Jha · Accepted Answer

Dear Student,
Here is the solution of your asked query:
Since A,B and C are in G P
⇒B2=AC     
(i)Also, X and Y are the arithmetic mean of A,B and B,C respectively
So we have;X=A+B2,Y=B+C2 Plugin the value of X and Y we have;1X+1Y=1A+B2+1B+C2 =2A+B+2B+C=2B+C+2A+BA+BB+C=2B+2C+2A+2BAB+AC+B2+BC=4B+2A+2CAB+B2+B2+BC       {using (i)}=4B+2A+2CAB+2B2+BC=22B+A+CB2B+A+C=2BTherefore, 1X+1Y=2BHence proved
Regards

A, B, C are in G.P and A, X, B, Y, C are A.P prove 1 X + 1 Y = 2 B .

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