mcTY-apgp This unit introduces sequences and series, and gives some simple examples of each. It also explores particular types of sequence known. Given the first term and the common difference of an arithmetic sequence find the recursive formula and the three terms in the sequence after the last one given. SOLUTION. The sequence is arithmetic with first term a1 = 50 and common difference The nth term of an arithmetic sequence with first term a1 and common.

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Some of the special patterns can be determined using your knowledge of sequences. In this module, you will learn about arithmetic sequences and series. Sequence: a list of numbers in a specific order. 1, 3, 4, 7, 10, • Term: each number in a sequence. Sequence. Terms. Notes Arithmetic Sequences and . Students will be able to understand how the common difference leads to the next term of an arithmetic sequence, the explicit form for an Arithmetic sequence.

Have you encountered numbers in sequence? Were you not puzzled as to how these numbers are arranged? Some numbers are arranged following special patterns. Some of the special patterns can be determined using your knowledge of sequences. In this module, you will learn about arithmetic sequences and series.

Do you understand the solutions to the examples presented? If you do, you may continue reading this module. If you do not, read the discussion and try answering the problems again. Let us try some more problems. Let us analyze the problem. This means that after 3 months, his wage will be P After 6 months, his wage will be P This means that in 5 years, his wage will increase by P5.

Therefore, we have the sequence 30, 35, 40, 45, 50,. You are asked for the 20th term in the given arithmetic sequence. Compare your answers with those found in the Answer Key on page If your answers are wrong, read the parts which are not clear to you again. If you got them right, congratulations.

You did a very good job. Try solving the next problem. How much would she have saved by the end of the month? What are the values given in the problem? We have 5, 5. What do these numbers form?

The numbers form an arithmetic sequence. You are asked for the amount that Analiza would be able to save by 30 September. This is equivalent to the 30th term of the arithmetic sequence 5, 5. Use the 5 steps in applying the formula for finding the nth term in an arithmetic sequence to solve the following.

Find the 28th term in an arithmetic sequence if the 1st term is 20 and the common difference is —5. What is the 17th term in the arithmetic sequence 7, 7. A stack of bricks has 61 bricks in the bottom layer, 58 bricks in the 2nd layer, 55 bricks in the 3rd layer, and 10 bricks in the last layer.

How many bricks are there in the 11th layer? The seats in a theater are arranged so that there are 70 seats in the 1st row, 72 seats in the 2nd row and so on for 30 rows altogether. How many seats are there in the last row? Once a month, a man puts some money in a cookie jar. During the 1st month he has P How much money was placed in the jar during the last month of the 4th year? Compare your answers with those found in the Answer Key on pages 39 and Continue reading this module.

If you got only 4 or below, read Lesson 2 again, then try solving the exercises given again. You will also learn how to apply that process in your daily life. From the series of numbers, find out the following: 1. Is the series an arithmentic sequence? Do the numbers in the series have a common difference? If yes, what is the common difference or d? What is the first term or a1? Before answering Question 1, first determine if there is a common difference.

Let us compute for the common difference by finding the difference between two consecutive numbers. Therefore, the numbers follow an arithmetic sequence. Now, the first term is This is a1. We can now solve for the number of soft drinks that Jaine-Anne can sell in 10 days.

Each of these sums is called an arithmetic series. An arithmetic series is the sum of the 1st n terms of an arithmetic sequence. In the example above, 90, , , , and so on is an example of an arithmetic series. Do you want to discover a formula for finding the sum of the 1st n terms of an arithmetic sequence? Earlier, we added the terms in the arithmetic sequence and we came up with the following sums: 90, , , , Let us write S to replace the sum.

S2 means the sum of the 1st two terms in an arithmetic sequence. How do you write the sum of the 1st three terms? S3 is the sum of the 1st three terms in an arithmetic sequence.

How do you write the sum of the 1st four terms? STEP 2 Determine the 1st term and the common difference in the given sequence. STEP 3 Find the formula for finding the unknown sum. Are your answers the same as those written below?

Suppose an is not given. Let us see how we can do this. When do we use Equation 2? Equation 2 is used when the 1st term, a1 and the nth term, an are given.

When do we use Equation 3? Equation 3 is used when the 1st term a1 and the common difference, d are given. If you cannot follow anymore, read the previous discussion again. In Example 1, the 1st term and the common difference are given so we can use Equation 3. To solve this, follow the steps below: STEP 1 Determine what the unknown is and write its arithmetic notation. In the given problem you are asked to determine the 1st 32 terms in the arithmetic sequence.

Thus, you are to solve for S STEP 2 Find out what the given facts in the problem are. STEP 3 Determine which formula to use, given the first term, a1 and the common difference, d. Remember, there are two equations or formulas for finding the sum of the first n terms. STEP 1 Determine what the given facts are. The 1st and last terms of the sequence are given. In the given problem, you are asked for the 1st 21 terms in the arithmetic sequence. STEP 3 Decide which formula to use for finding the unknown sum given the 1st and last terms.

How much money will she have at the end of 12 years? Here is my analysis. The wife saves P10, during the 1st year, P10, during the 2nd year, P10, during the 3rd year and so on until the 12th year.

The 1st term and the common difference of the sequence are given. STEP 3 Decide which formula to use for finding the unknown sum given the 1st term and the common difference. We are given the 1st term and the common difference, so we will use Equation 3.

Try solving the following by yourself. He sets aside P On the 8th week, he has saved P How much is the pair of shoes? Analyze the problem. The values given in the problem can be related to an arithmetic sequence because the savings are increased constantly. This means that there is a common difference. STEP 1 Determine the given facts. The 1st and the last terms of the sequence are given.

In the given problem, you are being asked for the sum of the 1st eight terms in the arithmetic sequence. STEP 3 Decide which formula should be used in finding the unknown sum given the 1st and last terms. Compare your answers with those given in the Answer Key on pages STEP 1 Determine what the given facts in the problem are.

If the 1st and last terms are given, use Equation 2. To determine the amount of money you will have saved after ten days, let us first analyze the problem. In this case: How much money will you save after 10 days? Third, solve the problem.

To solve the problem, present the information in tabular form as in: D ay Amount Saved 1 P Thsi si hte common edfrnice bw eten hte 4 Note, too, that you added P5.

Now, can you try working on the next problem by yourself? Every day he passes by the gasoline station to download diesel gas for his jeepney. With the increasing prices of gasoline and diesel, he wants to monitor the number of liters of diesel he uses every day. So, every day, he takes note of the number of liters of diesel his jeep uses up. On Monday, his jeep used up 7 liters of diesel gas.

On Tuesday, it used up 7. On Wednesday, he bought 8 liters of diesel gas. On Thursday, it used up 8. How many liters of diesel gas do you think will he download on Friday? What about on Saturday? It will be helpful to you if you will first determine the following information: An arithmetic sequence or arithmetic progression is a set or series of numbers following a certain pattern depending on the common difference.

The common difference is constant or a fixed number. Any series of numbers that do not have a common difference is not an arithmetic sequence or progression. Here are some more examples of arithmetic sequences.

If the given is an arithmetic sequence, write S on the space provided. If not, write N. You can proceed to the next lesson. If you got only 6 or less, read Lesson 1 again and after doing that, try to solve the exercises given above again. Consider the following set of numbers that represent the number of meters you cover while jogging each day: This represents an arithmetic sequence.

Each number in an arithmetic sequence is called a term. The 1st number is called the 1st term, the 2nd, the 2nd term, and so on. The 1st term in the sequence above is The 2nd term is The 3rd term is and so on. If we use a letter, like the letter a to replace any term in the sequence, we need to indicate whether it is the first, the second, or the third term in the sequence. This is done by writing a number at the lower right of the letter.

This number is called a subscript. In the example, a1 is the first term in the sequence. The symbol for the 2nd term is a2.

What about the symbols for the following terms? The third and fourth terms have been done for you. Remember our previous example on the number of meters covered when jogging? We can now represent those numbers as terms in a series. The numbers can be represented as: Thus, we have: Then by substituting 5 to d, we get: Substituting to a1, we get the following number sentence for a2: Thus we have: So we have: We will add 5 five times to the 1st term.

So the 6th term is: How many times will you add 5 to the 1st term? Give your answer by filling in the blanks below. Now, what have you noticed about our number sentences for the previous terms? Let us examine them closely. Compare this with the order of the terms. But you are multiplying d five times. This is also known as the formula for finding the nth term in an arithmetic sequence. Exercise Find the following nth terms in an arithmetic sequence with 25 as the first term and 3 as the common difference d.

Numbers 1 and 2 have been done for you. What is the 12th term in the given sequence? Follow the steps below to solve this problem. STEP 1 Find the common difference. STEP 2 Determine the 1st term in the given arithmetic sequence. The 1st term in the given arithmetic sequence is 2. STEP 3 Find the symbol for the unknown term in the sequence. You are asked for the 12th term in the given arithmetic sequence. Thus, the symbol for the unknown term is a STEP 4 Write the equation or the number sentence for the unknown term in the sequence.

The equation for a12 is: What do we do? We follow the same steps. But we should recall how to add and multiply signed numbers. Consider the following examples. Following the steps, we will have: The 1st term in the given arithmetic sequence is 5.

You are asked for the 15th term in the given arithmetic sequence. Thus, we solve for a STEP 4 Write the equation for the unknown term in the sequence. The equation for a15 is: The 15th term of the sequence 5, 3, 1, —1, —3, —5,. Now, try doing the next example by yourself. Just follow the steps. You are asked for the 17th term in the given arithmetic sequence.

The equation for a17 is: STEP 5 Substitute the values in the equation and solve for the result. Compare your answers with those given in the Answer Key on page Now, suppose you are tired of using the steps in solving for the nth term, what other method can you use? Remember the formula: Solve the following problem using the formula. The unknown in the problem is the 20th term in the sequence. This is represented as a Do you understand the solutions to the examples presented? If you do, you may continue reading this module.

If you do not, read the discussion and try answering the problems again. Let us try some more problems. Let us analyze the problem. The following are the given facts in the problem: This means that after 3 months, his wage will be P After 6 months, his wage will be P This means that in 5 years, his wage will increase by P5. Therefore, we have the sequence 30, 35, 40, 45, 50,. You are asked for the 20th term in the given arithmetic sequence. The equation for a20 is: Compare your answers with those found in the Answer Key on page If your answers are wrong, read the parts which are not clear to you again.

If you got them right, congratulations. You did a very good job. Try solving the next problem. How much would she have saved by the end of the month? What are the values given in the problem? We have 5, 5. What do these numbers form?

The numbers form an arithmetic sequence. You are asked for the amount that Analiza would be able to save by 30 September. This is equivalent to the 30th term of the arithmetic sequence 5, 5. We say that: Are your answers the same as those written below? Suppose an is not given. Let us see how we can do this. Using this in Equation 2, we have: When do we use Equation 2? Equation 2 is used when the 1st term, a1 and the nth term, an are given.

When do we use Equation 3? Equation 3 is used when the 1st term a1 and the common difference, d are given. If you cannot follow anymore, read the previous discussion again. In Example 1, the 1st term and the common difference are given so we can use Equation 3. From Step 3 we then have: To solve this, follow the steps below: STEP 1 Determine what the unknown is and write its arithmetic notation. In the given problem you are asked to determine the 1st 32 terms in the arithmetic sequence.

Thus, you are to solve for S STEP 2 Find out what the given facts in the problem are. The problem has the following given facts: STEP 3 Determine which formula to use, given the first term, a1 and the common difference, d.

Remember, there are two equations or formulas for finding the sum of the first n terms. The first formula: STEP 1 Determine what the given facts are. The 1st and last terms of the sequence are given.

In the given problem, you are asked for the 1st 21 terms in the arithmetic sequence. STEP 3 Decide which formula to use for finding the unknown sum given the 1st and last terms. From Step 3, we have: How much money will she have at the end of 12 years?

Here is my analysis. The wife saves P10, during the 1st year, P10, during the 2nd year, P10, during the 3rd year and so on until the 12th year.

The 1st term and the common difference of the sequence are given. STEP 3 Decide which formula to use for finding the unknown sum given the 1st term and the common difference. We are given the 1st term and the common difference, so we will use Equation 3.

From Step 3, we will have: Try solving the following by yourself. He sets aside P On the 8th week, he has saved P How much is the pair of shoes?

Analyze the problem. The values given in the problem can be related to an arithmetic sequence because the savings are increased constantly.

This means that there is a common difference. STEP 1 Determine the given facts. The 1st and the last terms of the sequence are given. In the given problem, you are being asked for the sum of the 1st eight terms in the arithmetic sequence. STEP 3 Decide which formula should be used in finding the unknown sum given the 1st and last terms. Compare your answers with those given in the Answer Key on pages STEP 1 Determine what the given facts in the problem are.

If the 1st and last terms are given, use Equation 2. If the 1st term and the common difference are given, use Equation 3. STEP 2 Determine what is being asked for in the problem and write this in arithmetic notation.

STEP 3 Decide which formula to use in finding the unknown sum based on the given facts or information. STEP 4 Substitute the given values in the formula and solve for the unknown. Use the four steps you just learned in finding the sum of the 1st n terms in an arithmetic sequence to solve the following: What is the sum of the 1st 11 terms in an arithmetic sequence whose 1st term is —40 and the 11th term is —73?

Use the formula Equation 2 or Equation 3 for finding the sum of the 1st n terms in an arithmetic sequence to solve the following problems.

A stack of bricks has 61 bricks in the bottom layer, 58 bricks in the 2nd layer, 55 bricks in the 3rd layer and 10 bricks in the last or 18th layer. How many bricks are there in all? How many seats in all are there in the theater? Once a month, a man puts money into a cookie jar. How much money had he put in the jar at the end of 4 years? Compare your answers with those given in the Answer Key on pages 40 and If you get a score below 4, read Lesson 3 again.

Review the parts you did not understand very well. Each term in the sequence, except the 1st one, is obtained by adding the common difference to the previous term. We follow the following steps in using the equation above: STEP 5 Substitute the values in the equation and solve for the unknown.

We follow the steps below in using the equations above: STEP 1 Determine the given facts in the problem. STEP 2 Determine what is being asked for in the problem and write in arithmetic notation. Let us check if you understood the topics discussed in this module. Answer the exercises given below. Determine if the given sequence is an arithmetic sequence or not.

Write Yes if it is and No if it is not. Use the formula for finding the nth term of the following arithmetic sequences. Use the appropriate formula in finding the sum of the 1st n terms of an arithmetic sequence given the following: Lydia is preparing for a coming marathon. She plans to run for seven days. How many kilometers will she have to cover on the last day? In a potato race, the first and last potatoes are 5 m and 15 m away, respectively, from the starting line and the rest are equally spaced 1 m away from each other.

What is the total distance traveled by a runner who brings them one at a time to the finishing line? For finishing a certain job, Pepe earns P How much will Pepe earn in 5 days? If your score falls between 14 — 15, congratulations. You did great. You really understood the topics discussed in this module. However, if you scored below 14, look at the descriptive ratings given below: Just review the parts of the module you did not understand very well.

Review the parts you did not understand and solve other exercises similar to those. Yes 2. Yes 5. Yes B. Lesson 2 Exercise page 13 3. Step 2: Step 1: Lesson 3 Exercise pages 30 — 31 Step 1: S8 Step 3: What Have You Learned? Yes 4. So, the number of terms in the series is Phoenix Publishing House, Inc. Reprinted Capitulo, F. A Simplified Approach. National Bookstore, Related Papers. By suraj sahoo. Exploring Progressions: A Collection of Problems.

By Konstantine Zelator. Real Sequences and Series. By Adeshina Adekunle. Chapter 4 Resource Masters. By dddddd d. Study Guide and Intervention Workbook. Remember me on this computer. Enter the email address you signed up with and we'll email you a reset link. Need an account?