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Magazine Chapter 2: Problem 3
What are the terms \(a_{0}, a_{1}, a_{2},\) and \(a_{3}\) of the sequence \(\left\\{a_{n}\right\\}\) where \(a_{n}\) equals $$ \begin{array}{ll}{\text { a) } 2^{n}+1 ?} & {\text { b) }(n+1)^{n+1} ?} \\\ {\text { c) }\lfloor n / 2\rfloor ?} & {\text { d) } \lfloor n / 2\rfloor+\lceil n / 2\rceil ?}\end{array} $$
Short Answer
Expert verified
a) 2, 3, 5, 9; b) 1, 2, 9, 64; c) 0, 0, 1, 1; d) 0, 1, 2, 3.
Step by step solution
01
- Evaluate the given sequences for a鈧
For each sequence, substitute into the formula and calculate the term for = 0.
02
Step 1a - Evaluate 2鈦 + 1 for n = 0
For = 0, = 2鈦 + 1 = 1 + 1 = 2, so 鈧 = 2.
03
Step 1b - Evaluate (n + 1)鈦 for n = 0
For = 0, = (0 + 1)鈦 = 1鈦 = 1, so 鈧 = 1.
04
Step 1c - Evaluate 鈱妌/2鈱 for n = 0
For = 0, = 鈱0/2鈱 = 鈱0鈱 = 0, so 鈧 = 0.
05
Step 1d - Evaluate 鈱妌/2鈱 + 鈱坣/2鈱 for n = 0
For = 0, = 鈱0/2鈱 + 鈱0/2鈱 = 0 + 0 = 0, so 鈧 = 0.
06
- Evaluate the given sequences for a鈧
For each sequence, substitute into the formula and calculate the term for = 1.
07
Step 2a - Evaluate 2鈦 + 1 for n = 1
For = 1, = 2鹿 + 1 = 2 + 1 = 3, so 鈧 = 3.
08
Step 2b - Evaluate (n + 1)鈦 for n = 1
For = 1, = (1 + 1)鹿 = 2鹿 = 2, so 鈧 = 2.
09
Step 2c - Evaluate 鈱妌/2鈱 for n = 1
For = 1, = 鈱1/2鈱 = 鈱0.5鈱 = 0, so 鈧 = 0.
10
Step 2d - Evaluate 鈱妌/2鈱 + 鈱坣/2鈱 for n = 1
For = 1, = 鈱1/2鈱 + 鈱1/2鈱 = 0 + 1 = 1, so 鈧 = 1.
11
- Evaluate the given sequences for a鈧
For each sequence, substitute into the formula and calculate the term for = 2.
12
Step 3a - Evaluate 2鈦 + 1 for n = 2
For = 2, = 2虏 + 1 = 4 + 1 = 5, so 鈧 = 5.
13
Step 3b - Evaluate (n + 1)鈦 for n = 2
For = 2, = (2 + 1)虏 = 3虏 = 9, so 鈧 = 9.
14
Step 3c - Evaluate 鈱妌/2鈱 for n = 2
For = 2, = 鈱2/2鈱 = 鈱1鈱 = 1, so 鈧 = 1.
15
Step 3d - Evaluate 鈱妌/2鈱 + 鈱坣/2鈱 for n = 2
For = 2, = 鈱2/2鈱 + 鈱2/2鈱 = 1 + 1 = 2, so 鈧 = 2.
16
- Evaluate the given sequences for a鈧
For each sequence, substitute into the formula and calculate the term for = 3.
17
Step 4a - Evaluate 2鈦 + 1 for n = 3
For = 3, = 2鲁 + 1 = 8 + 1 = 9, so 鈧 = 9.
18
Step 4b - Evaluate (n + 1)鈦 for n = 3
For = 3, = (3 + 1)鲁 = 4鲁 = 64, so 鈧 = 64.
19
Step 4c - Evaluate 鈱妌/2鈱 for n = 3
For = 3, = 鈱3/2鈱 = 鈱1.5鈱 = 1, so 鈧 = 1.
20
Step 4d - Evaluate 鈱妌/2鈱 + 鈱坣/2鈱 for n = 3
For = 3, = 鈱3/2鈱 + 鈱3/2鈱 = 1 + 2 = 3, so 鈧 = 3.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Sequence Evaluation
Sequence evaluation is an essential concept in discrete mathematics. It involves determining the terms of a sequence based on a given formula. In the original exercise, we evaluated four different sequences at different positions (n = 0, 1, 2, 3). To find any term in a sequence, simply replace the variable in the formula with the desired term's position number (n). Let's go through a quick example:
Consider the sequence defined as \(a_n = 2^n + 1\). To find \(a_2\), substitute \(n = 2\):
Consider the sequence defined as \(a_n = 2^n + 1\). To find \(a_2\), substitute \(n = 2\):
- \(a_2 = 2^2 + 1 = 4 + 1 = 5\)
Floor and Ceiling Functions
Floor and ceiling functions are used to round numbers to the nearest integers. The floor function \(\lfloor x \rfloor\) rounds down to the nearest integer less than or equal to \(x\), while the ceiling function \(\lceil x \rceil\) rounds up to the nearest integer greater than or equal to \(x\). These functions are particularly useful in discrete mathematics where exact integer values are often required.
Let's see an example from the exercise:
To find \(\lfloor 1/2 \rfloor\), we note that \(1/2\) or 0.5 is between 0 and 1:
Let's see an example from the exercise:
To find \(\lfloor 1/2 \rfloor\), we note that \(1/2\) or 0.5 is between 0 and 1:
- Hence, \(\lfloor 1/2 \rfloor = 0\)
- Hence, \(\lceil 1/2 \rceil = 1\)
Exponential Functions
Exponential functions are expressed in the form \(a^n\), where \(a\) is a constant base and \(n\) is an exponent. These functions grow very quickly and are commonly found in various discrete mathematics problems.
In the given exercise, the first sequence formula \(a_n = 2^n + 1\) is an exponential function. To evaluate exponential functions, simply replace \(n\) with any integer value:
For example:
In the given exercise, the first sequence formula \(a_n = 2^n + 1\) is an exponential function. To evaluate exponential functions, simply replace \(n\) with any integer value:
For example:
- For \(n = 3\), \(a_3 = 2^3 + 1 = 8 + 1 = 9\)
Discrete Mathematics Problems
Discrete mathematics focuses on studying mathematical structures that are fundamentally discrete rather than continuous. This means that it deals with countable, distinct elements.
Problems in discrete mathematics often involve sequences, series, logic, combinatorics, graph theory, and number theory. Sequence evaluation, like in our example, is a common type of discrete mathematics problem. It helps build a strong foundation for more advanced topics.
Key techniques in solving these problems include:
Problems in discrete mathematics often involve sequences, series, logic, combinatorics, graph theory, and number theory. Sequence evaluation, like in our example, is a common type of discrete mathematics problem. It helps build a strong foundation for more advanced topics.
Key techniques in solving these problems include:
- Breaking down complex expressions into simple steps
- Using functions like floor and ceiling for precision
- Applying mathematical concepts to real-world problems
