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# In an AP of 10 numbers and a HP of 10 numbers, the first term and last term are the same and equal to 2 and 3. Find the value of product of fourth term of the AP and the seventh term of the HP.

$\begin{array}{1 1} 2 \\ 3 \\ 5 \\ 6 \end{array}$

Explanation : Let the AP = $a_{1},a_{2},a_{3}--------a_{10}$
$HP\;=h_{1},h_{2},h_{3}--------h_{10}$
$a_{1}=h_{1}=2=a$
$a_{10}=h_{10}=3$
$a_{10}=a+9d=2+9d=3$
$d=\frac{1}{9}$
$h_{10}=3\quad\;\frac{1}{3}=\frac{1}{2}+9D$
$D=\frac{-1}{54}$
$a_{4}=a+3d=2+\frac{3}{9}=\frac{7}{3}$
$h_{7}\qquad\;\frac{1}{h_{1}}=\frac{1}{h_{1}}+6D=\frac{1}{2}-\frac{1}{9}=\frac{7}{18}$
$h_{7}=\frac{18}{7}$
$a_{4}\;h_{7}=\frac{7}{3}*\frac{18}{7}=6.$