Question 14

Explanation

Given the recursive formula $a_{n+1} = 4a_n + 1$ and $a_4 = 85$.

We want to find the divisibility pattern of $a_n$ by $5$. Let's calculate the first few terms modulo $5$.

First, let's find $a_3, a_2,$ and $a_1$:

Continue backwards to $a_2$:

Continue backwards to $a_1$:

The pattern of remainders when $a_n$ is divided by $5$ is:

• $a_1 \equiv 1$
• $a_2 \equiv 0$ (divisible by 5)
• $a_3 \equiv 1$
• $a_4 \equiv 0$ (divisible by 5)

It appears that $a_n$ is divisible by $5$ if and only if $n$ is an even number.

Let's check the available options:

• $1$ (Odd)
• $29$ (Odd)
• $73$ (Odd)
• $364$ (Even)
• $473$ (Odd)

The only even number among the choices is $364$.

Thus, the value of $n$ such that $a_n$ is divisible by $5$ is $364$.