Prove by contraposition that if n + 1 is an odd integer, then n is an even integer. Let {a_n} be a sequence that satisfies the recurrence relation: a_n = a_{n-1} - a_{n-2} for n =... Prove by contraposition that if n + 1 is an odd integer, then n is an even integer. Let {a_n} be a sequence that satisfies the recurrence relation: a_n = a_{n-1} - a_{n-2} for n = 2, 3, 4, ..., and suppose that a_0 = 3 and a_1 = 5. Find the value of a_2, a_3, and a_4. Show your calculation.
Understand the Problem
The question asks to prove a mathematical statement using contrapositive reasoning and to find specific values of a sequence defined by a recurrence relation. It involves logical reasoning and calculations based on the provided recurrence formula.
Answer
For question 20, the contrapositive is proved. For question 21, \( a_2 = 2 \), \( a_3 = -3 \), \( a_4 = -5 \).
Answer for screen readers
For question 20, the contrapositive statement is proven.
For question 21, the values are:
- ( a_2 = 2 )
- ( a_3 = -3 )
- ( a_4 = -5 )
Steps to Solve
- Prove the Contrapositive for Question 20
To prove the statement "if $n + 1$ is an odd integer, then $n$ is an even integer," we need to show its contrapositive: "if $n$ is an odd integer, then $n + 1$ is an even integer."
Let $n = 2k + 1$, where $k$ is an integer (this is the definition of an odd integer).
Now, calculate $n + 1$:
$$ n + 1 = (2k + 1) + 1 = 2k + 2 = 2(k + 1) $$
Since $k + 1$ is an integer, $n + 1$ is even. Thus, the contrapositive is proven.
- Set Up the Recurrence Relation for Question 21
The recurrence relation given is:
$$ a_n = a_{n-1} - a_{n-2} \quad \text{for } n = 2, 3, 4, \ldots $$
With initial conditions: ( a_0 = 3 ) and ( a_1 = 5 ).
- Calculate the Values Step by Step
First, find ( a_2 ):
$$ a_2 = a_1 - a_0 = 5 - 3 = 2 $$
Next, find ( a_3 ):
$$ a_3 = a_2 - a_1 = 2 - 5 = -3 $$
Finally, find ( a_4 ):
$$ a_4 = a_3 - a_2 = -3 - 2 = -5 $$
For question 20, the contrapositive statement is proven.
For question 21, the values are:
- ( a_2 = 2 )
- ( a_3 = -3 )
- ( a_4 = -5 )
More Information
In question 20, using contrapositive reasoning is a powerful tool in mathematical proofs, especially in logic and number theory. In question 21, recurrence relations are a common way to define sequences, and the process of calculating terms step-by-step is essential for understanding the behavior of such sequences.
Tips
- In question 20, a common mistake is misinterpreting odd and even definitions. Remember to represent odd integers as $2k + 1$ and even integers as $2k$.
- For question 21, forgetting the initial conditions can lead to an incorrect sequence. Always begin with the provided ( a_0 ) and ( a_1 ).
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