Given Vertices and Foci, we get $a = 5$ and $c = 4$.
$c= \sqrt {a^2 - b^2} \rightarrow 16 = \sqrt{25-b^2} \rightarrow b = \sqrt 9 = 3$
We therefore can write the equation of the ellipse as $\;\large\frac{x^2}{5^2}$$+\large\frac{y^2}{3^2}$$=1$
$\Rightarrow$ The equation of the ellipse is $\large\frac{x^2}{25}$$+\large\frac{y^2}{9}$$=1$