Question

Number of integral terms in the expansion of $\left( \sqrt{7} z + \frac{1}{6 \sqrt{z}} \right)^{824}$ is equal to ______.

Answer 1

Solution: The general term in the expansion is:

$t_{r+1} = \binom{824}{r} \left( \sqrt{7} z \right)^{824 - r} \left( \frac{1}{6 \sqrt{z}} \right)^r$

which simplifies to $z^{\frac{824 - r}{7} - \frac{r}{6}}$.

For an integral power, $r$ must be a multiple of 6.

Thus, $r = 0, 6, 12, \dots, 822$, giving 138 integral terms.