Question 51
Given that $\{a_n\}$ and $\{b_n\}$ are two sequences of integers defined by
$$
a_1 = 1,\ a_2 = 10,\ a_{n+1} = 2a_n + 3a_{n-1} \quad \text{for } n = 2, 3, 4, \dots,
$$
$$
b_1 = 1,\ b_2 = 8,\ b_{n+1} = 3b_n + 4b_{n-1} \quad \text{for } n = 2, 3, 4, \dots
$$
Prove that, besides the number ‘1’, no two numbers in the sequences are identical.
$$
a_1 = 1,\ a_2 = 10,\ a_{n+1} = 2a_n + 3a_{n-1} \quad \text{for } n = 2, 3, 4, \dots,
$$
$$
b_1 = 1,\ b_2 = 8,\ b_{n+1} = 3b_n + 4b_{n-1} \quad \text{for } n = 2, 3, 4, \dots
$$
Prove that, besides the number ‘1’, no two numbers in the sequences are identical.