Question 15

Explanation

We are asked to find a number that divides the difference between two consecutive terms, $a_{2025} - a_{2024}$.

To make it easier, let's find the pattern of the difference between consecutive terms by calculating the initial terms first.

We know $a_4 = 85$ and the formula $a_{n+1} = 4a_n + 1$. We can find the previous term by reversing the formula:

Finding the values of initial terms

• Find $a_3$:

  $$a_3 = \frac{a_4 - 1}{4} = \frac{85 - 1}{4} = \frac{84}{4} = 21$$

• Find $a_2$:

  $$a_2 = \frac{a_3 - 1}{4} = \frac{21 - 1}{4} = \frac{20}{4} = 5$$

• Find $a_1$:

  $$a_1 = \frac{a_2 - 1}{4} = \frac{5 - 1}{4} = \frac{4}{4} = 1$$

Observing the pattern of differences between terms

Let's calculate the difference between adjacent terms:

• $a_2 - a_1 = 5 - 1 = 4 = 4^1$
• $a_3 - a_2 = 21 - 5 = 16 = 4^2$
• $a_4 - a_3 = 85 - 21 = 64 = 4^3$

From the pattern above, we can see that the difference between the $(n+1)$-th and $n$-th term is a power of 4:

Determining the divisor

Based on the pattern we found, the difference $a_{2025} - a_{2024}$ is:

The value $4^{2024}$ is a power of 4. Therefore, this number is definitely divisible by 4.

Thus, the number that divides $a_{2025} - a_{2024}$ is $4$.