Triangle spiral

The hypotenuse of the $n$ th triangle becomes the base of the $(n + 1)$ st. Since the ratio of the hypotenuse- to base-length of the first triangle is $\frac{5}{4}$ , the sequence of hypotenuse lengths will be geometric with a first term of $5$ and ratio of $\frac{5}{4}$ . Thus the length of the $9$ th triangle's hypotenuse will be

$5 * (\frac{5}{4})^{9 - 1} = \dfrac{5^{9}}{4^{8}} = \dfrac{1953125}{65536}$ ,

giving us the solution $a + b = 1953125 + 65536 = \boxed{2018661}$ .

Did it but wasn't sure that there could be such a big anwsers on brilliant...

for 1st triangle hypotenuses will be 5, for 2nd triangle it will be 5 x(5/4)^1 for 3nd triangle it will be 5 x(5/4)^2 for 4nd triangle it will be 5 x(5/4)^3 similarly for 9 nd triangle it will be 5 x(5/4)^8= 1953125/65536 =>2018