OMM 2000 Problem 4

Let $a$ and $b$ be positive integers not divisible by $5$ . A sequence of integers is constructed as follows: the first term is $5$ , and every consequent term is obtained by multiplying its precedent by $a$ and adding $b$ . (For example, if $a = 2$ and $b = 4$ , the first three terms are $5,14,32$ .) What is the maximum possible number of primes that can occur before encoutering the first composite term?