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Q.
If $a _{1}, a _{2}, a _{3} \ldots$ and $b _{1}, b _{2}, b _{3} \ldots .$ are A.P. and $a_{1}=2, a_{10}=3, a_{1} b_{1}=1=a_{10} b_{10}$ then $a_{4} b_{4}$ is equal to
$a _{1}, a _{2}, a _{3} \ldots . A . P . ; a _{1}=2 ; a _{10}=3 ; d _{1}=\frac{1}{9}$
$b _{1}, b _{2}, b _{3}, \ldots$ A.P. $; b _{1}=\frac{1}{2} ; b _{10}=\frac{1}{3} ; d _{2}=\frac{-1}{54}$
[Using $a _{1} b _{1}=1= a _{10} b _{10} ; d _{1}\, \&\, d _{2}$ are common differences respectively]
$a _{4} \cdot b _{4} =\left(2+3 d _{1}\right)\left(\frac{1}{2}+3 d _{2}\right)$
$=\left(2+\frac{1}{3}\right)\left(\frac{1}{2}-\frac{1}{18}\right)$
$=\left(\frac{7}{3}\right)\left(\frac{8}{18}\right)=\frac{28}{27}$