prove that this is in A.P
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SEE THE AATACHMENT YOUR ANSWER IS THIS AND I THINK IT IS CORRECT
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If a, b, c is a harmonic progression, then we can say that a = 1/k, b = 1/(k+d) and c = 1/(k+2d). We can replace the expressions for a, b, c into the next three terms:
=> b+c/bc
=> (1/k+2d*1/2k+d)/(1/k+d+1/2k+d)
=> 1/2k+3d
Similarly,
=> c+a/ca
=> (1/k+d*1/k)/(1/k+d+1/k)
=> 1/2k+2d
=> a+b/ab
=> (1/k*1/k+d)/(1/k+d+1/2k+d)
=> 1/2k+d
We see the third, second, and first terms (in that order) follow a harmonic progression because the reciprocals of each make an arithmetic progression with common difference d.
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