Homework 6 Arithmetic Sequences Answer Key

[Eberardo Topher]

Inyesterday's lesson, we talked about arithmetic and geometric sequences and how to write recursive formulas. Today we will focus in on just arithmetic sequences. Our goal is for students to use their understanding of patterns and recursive formulas to create explicit formulas for arithmetic sequences. We also will draw students' attention to the fact that when graphed, arithmetic sequences form a straight line. Later this chapter, we will connect these explicit formulas to linear functions.

Begin by having students work through the entire Activity in groups. As you are helping students, make sure to point out ideas that they learned in the previous lesson. For example, you could ask students, "What's that called again when you add the same number over and over?" You should also ask a variety of different groups to write their solutions on the board as they are working through it.

After students complete all of the Activity, it's time to debrief. For the first 2 questions, we're using the information we learned yesterday to identify that we have an arithmetic sequence with a common difference of 3. For question 3, ask a student to explain what they noticed about their graph. When they mention that the points go in a straight line, explain and label that means the pattern is linear. For questions 4 and 5, we want to make sure that students understand why we are adding 3 employees 23 times, not 24 times. Ask the group who wrote their work on the board to explain this. You can even ask them why they didn't do 3(24) for 24 months. Lastly in question 5, you'll want to point out how the rule can be written more than one way, but they all simplify down to the same rule.

This unit is all about understanding linear functions and using them to model real world scenarios. Fluency in interpreting the parameters of linear functions is emphasized as well as setting up linear functions to model a variety of situations. Linear inequalities are also taught. The unit ends with a introduction to sequences with an emphasis on arithmetic.

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An arithmetic sequence is an ordered set of numbers that have a common difference between each consecutive term.

For example in the arithmetic sequence 3, 9, 15, 21, 27, the common difference is 6.

An arithmetic sequence can be known as an arithmetic progression. The difference between consecutive terms is an arithmetic sequence is always the same.

Arithmetic sequences are also known as linear sequences. If we represented an arithmetic sequence on a graph it would form a straight line as it goes up (or down) by the same amount each time. Linear means straight.

In order to continue an arithmetic series, you should be able to spot, or calculate, the term-to-term rule. This is done by subtracting two consecutive terms to find the common difference.

Arithmetic sequence is part of our series of lessons to support revision on sequences. You may find it helpful to start with the main sequences lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

In order to find missing numbers in an arithmetic sequence, we use the common difference. This can be useful when you are asked to find large terms in the sequence and you have been given a consecutive number to the term you are trying to calculate.

In order to generate an arithmetic sequence, we need to know the nth term.

The nth term is the name or rule that the sequence must follow to generate an ordered list of numbers.

4. Below are the first 3 terms of a pattern. The number of lines is represented by the sequence 4n+1 and the number of triangles is represented by the sequence 2n . How many lines are there in the term with 12 triangles?

This unit provides an opportunity to revisit representations of functions (including graphs, tables, and expressions) at the beginning of the Algebra 2 course, and also introduces the concept of sequences. Through many concrete examples, students learn to identify geometric and arithmetic sequences. Beginning with an invitation to describe sequences informally, students progress to writing terms of sequences arising from mathematical situations, using representations such as tables and graphs. They progress to using function notation to define sequences recursively and then explicitly for the \(n^\textth\) term. Throughout the unit, students learn that sequences are functions and that geometric and arithmetic sequences are examples of the exponential and linear functions they learned about in previous courses, defined on a subset of the integers. In the last part of the unit, students use sequences to model several situations represented in different ways. Finally, students encounter some situations where it makes sense to compute the sum of a finite sequence. A formula for such a sum is developed in a future unit.

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A sequence is a list of terms that have been ordered in a sequential manner and any sort of repetition is allowed. An arithmetic sequence is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. Arithmetic sequences worksheets help students build basic ideas on sequences and series in mathematics.

Sequences play an important role in our life. Practicing arithmetic sequences worksheets help us to predict and evaluate the outcome of a situation. Sequences are applicable if we look for a pattern that aids in obtaining the general term. Real-life examples include stacking cups, chairs, bowls, and pyramid-like patterns where objects are increasing or decreasing in a constant manner.

This set is also closed under vector addition. Consider from above and where and for some we have for , and consider the sum . Choose so that and . Then for and for so that for . The sum is therefore also a member of the set.

(d) We first check that the set of converging sequences is closed under scalar multiplication. Let be a member of this set, so that exists. Then for any there exists such that for . Now consider where is any scalar. If then so that it converges to the limit 0.

This means that exists and is equal to so that for any scalar and converging sequence the sequence is also in the set of converging sequences. Since converges both for and the set is therefore closed under scalar multiplication.

So for any we can choose such that for all . This means that exists and is equal to so that for any two converging sequences and the sequence is also in the set of converging sequences. The set is therefore closed under vector addition.

Since the set of converging sequences is closed under both vector addition and scalar multiplication and it is a subset of the vector space of infinite sequences, it is a subspace of that vector space.

(e) We first check that the set of arithmetic progressions is closed under scalar multiplication. Let be a member of this set, so that is a constant value for all . Then for we have for all . The sequence is therefore also an arithmetic progression, and the set is closed under scalar multiplication.

Since the set of arithmetic progressions is closed under both vector addition and scalar multiplication and it is a subset of the vector space of infinite sequences, it is a subspace of that vector space.

(f) The set of geometric progressions is not a subspace because it is not closed under vector addition: For example, suppose that (for which and ), and (for which and ). We then have . For we have (from the first element) and (from the second element). If is a geometric progression then the third element should be instead of the actual value of 13. So is not a geometrical progression for all geometric progressions and , and the set is not closed under vector addition.

Actually (b) is a special case of (a): (a) includes all sequences with an infinite number of zeroes, while (b) includes only such sequences where there are only zeroes (and no other number) past a certain point. So, for example, (a) would include a sequence where every millionth number was a zero, but such a sequence would not be a member of (b).

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