The product of the third and the eighth terms of a G.P is 243. If the fourth term is 3, find its seventh term.
Third term of G.P $= t_3 = ar^2$
Eighth term of G.P $= t_8 = ar^7$
$\therefore$ $ar^2 \times ar^7 = 243$ $\implies$ $a^2 r^9 = 243$ $\cdots$ (1)
Fourth term of G.P $= t_4 = ar^3 = 3$ $\cdots$ (2)
Now, equation (1) can be rewritten as
$ar^3 \times ar^3 \times r^3 = 243$ $\cdots$ (3)
In view of equation (2), equation (3) becomes
$3 \times 3 \times r^3 = 243$ $\implies$ $r^3 = \dfrac{243}{9} = 27$ $\implies$ $r = \sqrt[3]{27} = 3$
Substituting the value of r in equation (2) gives
$a \times 3^3 = 3$ $\implies$ $a = \dfrac{1}{9}$
$\therefore$ Seventh term $= t_7 = ar^6 = \dfrac{1}{9} \times 3^6 = 81$