수열 전체 문제

For the sequence $a_1 ,a_2 , \ldots ,a_n , \ldots$, assume that $a_1=1$, $a_2=1$, $a_3=1$, and for that each natural number $n$,

$$a_{n+3} = a_{n+2} + a_{n+1} + a_n$$

.

Prove that for each natural number $n$ with $n>1$, $a_n \leq 2^{n-2}$.

For the sequence $a_1, a_2, \ldots, a_n, \ldots$ , assume that $a_1 = 1$, $a_2 = 1$, and for each $n \in \mathbb{N}$, $a_{n+2} = a_{n+1} + 3a_n$. Determine which terms in this sequence are divisible by 4 and prove that your answer is correct.

The Lucas numbers

are a sequence of natural numbers $L_1, L_2, L_3, \ldots, L_n, \ldots$, which are defined recursively as follows:

• $L_1 = 1$ and $L_2 = 3$, and - For each natural number $n$, $L_{n+2} = L_{n+1} + L_n$.

List the first 10 Lucas numbers and the first ten Fibonacci numbers and then prove each of the following propositions. The Second Principle of Mathematical Induction may be needed to prove some of these propositions.

For each natural number $n$, $L_n = 2f_{n+1} - f_n$.

The Lucas numbers

are a sequence of natural numbers $L_1, L_2, L_3, \ldots, L_n, \ldots$, which are defined recursively as follows:

• $L_1 = 1$ and $L_2 = 3$, and - For each natural number $n$, $L_{n+2} = L_{n+1} + L_n$.

List the first 10 Lucas numbers and the first ten Fibonacci numbers and then prove each of the following propositions. The Second Principle of Mathematical Induction may be needed to prove some of these propositions.

For each $n \in \mathbb{N}$ with $n \geq 2$, $5f_n = L_{n - 1} + L_{n + 1}$.

The Lucas numbers

are a sequence of natural numbers $L_1, L_2, L_3, \ldots, L_n, \ldots$, which are defined recursively as follows:

• $L_1 = 1$ and $L_2 = 3$, and - For each natural number $n$, $L_{n+2} = L_{n+1} + L_n$.

List the first 10 Lucas numbers and the first ten Fibonacci numbers and then prove each of the following propositions. The Second Principle of Mathematical Induction may be needed to prove some of these propositions.

For each $n \in \mathbb{N}$ with $n \geq 3$, $L_n = f_{n + 2} - f_{n - 2}$.