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Question

If a,b,c are in AP or GP or HP, then a−bb−c is equal to

A
ba or 1 or bc
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B
ca or cbor1
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C
1 or ab or ac
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D
1 or ab or ca
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Solution

The correct options are
A ca or cbor1
B ba or 1 or bc
C 1 or ab or ac
D 1 or ab or ca
If a, b and c are in AP, then common difference d=cb=ba

abbc=1

Also, b=a+d and c=b+d

If a, b and c are in GP let the common ratio be r.
Then b=ar and c=br.

abbc=aarbbr=ab

Also c=ar2

abbc=aararar2=1r=arar2=bc

So, abbc=ab=bc

If a,b,c are in HP, then 1a,1b,1c are in AP.
Then, b=2aca+c

abbc=a2aca+c2aca+cc=a2acacc2=a(ac)c(ac)=ac


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