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Question

# If the (m+1)th,(n+1)th&(r+1)th terms of an AP are in GP & m,n,r are in HP, then the ratio of the common difference to the first term of the AP is

A
1n
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B
2n
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C
2n
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D
none of these
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Solution

## The correct option is B −2nGiven m,n,r are in H.P. ⇒mr=n(m+r)2..........(i)Given, (m+1)th,(n+1)th&(r+1)th are terms of an APTm+1=a+mdTn+1=a+ndTr+1=a+rdAlso, given these are in GP.⇒a+rda+nd=a+nda+md⇒1+r(d/a)1+n(d/a)=1+n(d/a)1+m(d/a)Let da=x⇒(1+nx)2=(1+rx)(1+mx)⇒(n2−mr)x2+(2n−r−m)x=0⇒x=0 or x=−(2n−r−mn2−mr)=⎛⎜ ⎜ ⎜⎝2n−r−mn(m+r)2−n2⎞⎟ ⎟ ⎟⎠ (by (i))⇒x=−2n

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