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Let a(1), a(2),"…..",a(10) be in A.P. an...

Let `a_(1), a_(2),"…..",a_(10)` be in A.P. and `h_(1) ,h_(2),"…….",h_(10)` be in H.P.
If `a_(1) = h_(1) = 2 ` and `a_(10) = h_(10) = 3` , then `a_(4)h_(7)` is `:`

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` therefore a_(1),a_(2),a_(3),"...."a_(10)` are in AP.
If d be the common difference, then `d=(a_(10)-a_(1))/(9)=(3-2)/(9)=(1)/(9)`
`therefore a_(4)=a_(1)+3d=2+(3)/(9)=2+(1)/(3)=(7)/(3) "……..(i)"`
and given ` h_(1),h_(2),h_(3),"...."h_(10)` are in HP.
If D be common difference of corresponding AP.
Then, ` D=((1)/h_(10)-(1)/h_(1))/(9)=((1)/(3)-(1)/(2))/(9)=-(1)/(54)`
` therefore (1)/h_(7)=(1)/h_(1)+6D=(1)/(2)-(6)/(54)=(1)/(2)-(1)/(9)=(7)/(18) implies h_(7)=(18)/(7)`
Hence, ` a_(4)*h_(7)=(7)/(3)xx(18)/(7)=6`
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