Problem 20 For the following exercises, wri... [FREE SOLUTION]

Chapter 11: Problem 20

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{ll}{4\left(n^{2}-2\right)} & {\text { if } n \leq 3 \text { or } n>6} \\ {\frac{n^{2}-2}{4}} & {\text { if } 3

Short Answer

Expert verified

The first eight terms are -4, 8, 28, 3.5, 5.75, 8.5, 188, 248.

Step by step solution

Understand the Sequence

This sequence is defined piecewise, meaning it has different expressions for different values of $ n $. For $ n \leq 3 $ or $ n > 6 $, the sequence is defined as $ 4(n^2 - 2) $. For $ 3 < n \leq 6 $, it is defined as $ \frac{n^2 - 2}{4} $. We need to find the first eight terms for $ n = 1, 2, 3, 4, 5, 6, 7, 8 $.

Calculate Terms for $ n \leq 3 $

Evaluate the expression $ 4(n^2 - 2) $ for $ n = 1, 2, 3 $.- For $ n = 1 $: \[ a_1 = 4(1^2 - 2) = 4(1 - 2) = 4(-1) = -4 \]- For $ n = 2 $: \[ a_2 = 4(2^2 - 2) = 4(4 - 2) = 4(2) = 8 \]- For $ n = 3 $: \[ a_3 = 4(3^2 - 2) = 4(9 - 2) = 4(7) = 28 \]

Calculate Terms for $ 3 < n \leq 6 $

Evaluate the expression $ \frac{n^2 - 2}{4} $ for $ n = 4, 5, 6 $.- For $ n = 4 $: \[ a_4 = \frac{4^2 - 2}{4} = \frac{16 - 2}{4} = \frac{14}{4} = 3.5 \]- For $ n = 5 $: \[ a_5 = \frac{5^2 - 2}{4} = \frac{25 - 2}{4} = \frac{23}{4} = 5.75 \]- For $ n = 6 $: \[ a_6 = \frac{6^2 - 2}{4} = \frac{36 - 2}{4} = \frac{34}{4} = 8.5 \]

Calculate Terms for $ n > 6 $

Evaluate the expression $ 4(n^2 - 2) $ for $ n = 7, 8 $.- For $ n = 7 $: \[ a_7 = 4(7^2 - 2) = 4(49 - 2) = 4(47) = 188 \]- For $ n = 8 $: \[ a_8 = 4(8^2 - 2) = 4(64 - 2) = 4(62) = 248 \]

Write the First Eight Terms

Compile the calculated values into the sequence:- First Term ($ n = 1 $): -4- Second Term ($ n = 2 $): 8- Third Term ($ n = 3 $): 28- Fourth Term ($ n = 4 $): 3.5- Fifth Term ($ n = 5 $): 5.75- Sixth Term ($ n = 6 $): 8.5- Seventh Term ($ n = 7 $): 188- Eighth Term ($ n = 8 $): 248

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

sequence terms calculation

A piecewise sequence is a mathematical sequence where the rule for determining its terms varies depending on the value of the variable, in this case, $ n $. Calculating the terms of such a sequence involves understanding and applying different expressions to different intervals of $ n $. In the given example, the sequence is defined for specific ranges:

For $ n \leq 3 $ or $ n > 6 $, the expression $ 4(n^2 - 2) $ is used.
For $ 3 < n \leq 6 $, the expression $ \frac{n^2 - 2}{4} $ is used.

To find the first eight terms, we evaluate these expressions individually by substituting $ n = 1, 2, 3, 4, 5, 6, 7, $ and $ 8 $. This process ensures that each term is computed according to its designated rule, forming the piecewise sequence.

mathematical expressions

Mathematical expressions are combinations of numbers, variables, and operators (such as addition, subtraction, multiplication, and division) that represent a particular value or relationship. They often define the rules for calculating terms in sequences, like in our piecewise sequence example.In our sequence:

The expression $ 4(n^2 - 2) $ implies a quadratic relationship with $ n $ altered by a constant difference and a multiplication factor of 4. This drastically changes the term value as $ n $ increases for specific ranges.
The expression $ \frac{n^2 - 2}{4} $ also represents a quadratic relationship moderated by division by 4, which gently adjusts the term value for another specific range of $ n $.

By substituting values of $ n $ into these expressions, we calculate each term individually, maintaining the sequence鈥檚 integrity and its adherence to defined mathematical rules.

step by step solution

Following a step by step solution is crucial for understanding complex mathematical concepts. For piecewise sequences, it helps to break down the sequences based on the defined intervals of $ n $, ensuring each step accounts for the respective expression to use.Here's how the solution was approached:

**Step 1:** Recognize the sequence is piecewise and note the expressions correspond to the values of $ n $.
**Step 2:** For $ n \leq 3 $, evaluate $ 4(n^2 - 2) $, calculating each term by substituting the values for $ n = 1, 2, 3 $.
**Step 3:** For $ 3 < n \leq 6 $, evaluate $ \frac{n^2 - 2}{4} $ for $ n = 4, 5, 6 $.
**Step 4:** For $ n > 6 $, return to $ 4(n^2 - 2) $ and compute terms for $ n = 7, 8 $.

This structured approach helps streamline calculations, and clarifies how the inputs ($ n $) change the outcome based on the piecewise definition. This ensures accuracy and aids in learning how to manage different parts of a sequence effectively.

91影视

For the following exercises, write the first eight terms of the piecewise sequence. $$ a_{n}=\left\\{\begin{array}{ll}{4\left(n^{2}-2\right)} & {\text { if } n \leq 3 \text { or } n>6} \\ {\frac{n^{2}-2}{4}} & {\text { if } 3

Short Answer

Step by step solution

Understand the Sequence

Calculate Terms for \( n \leq 3 \)

Calculate Terms for \( 3 < n \leq 6 \)

Calculate Terms for \( n > 6 \)

Write the First Eight Terms

Key Concepts

sequence terms calculation

mathematical expressions

step by step solution

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