If the number of integral terms in the expansion of (3(1)/(2) +...

${\left(3\dfrac{1}{2} + 5\dfrac{1}{8}\right)}^{n}$

General term ${T}_{r + 1} = Cr n {\left(3\dfrac{1}{2}\right)}^{n - r}{\left(5\dfrac{1}{8}\right)}^{r}$

${T}_{r + 1} = Cr n 3\dfrac{n - r}{2}5\dfrac{r}{8}$.

$\dfrac{n - r}{2}, \dfrac{r}{8}$ must be integer.

$\dfrac{n - r}{2} \in I ⇒ n - r = 0, 2, 4, 6, 8 . . . . . . .$

$\dfrac{r}{8} \in I ⇒ r = 0, 8, 16, 24, 32 . . . .$

For rational term common powers$= 0, 8, 16, 24 . . . . . . .$

Number of integral term$= 33$

${T}_{s} = a + \left(s - 1\right)d ⇒ {T}_{33} = 0 + \left(33 - 1\right)8 = 256$.

Value of $n = 256$.

We can cross check to verify answer.