Topic 5 “Modeling with Linear and Quadratic Functions” 5-1 Arithmetic Sequences & Series

Arithmetic Sequences Every day a radio station asks a question for a prize of $150. If the 5th caller does not answer correctly, the prize money increased by $150 each day until someone correctly answers their question.

Arithmetic Sequences Make a list of the prize amounts for a week (Mon - Fri) if the contest starts on Monday and no one answers correctly all week. Monday : $150 Tuesday: $300 Wednesday: $450 Thursday:$600 Friday:$750 These prize amounts form a sequence, more specifically each amount is a term in an arithmetic sequence. To find the next term we just add $150.

What is….? (Definitions) Sequence: a list of numbers in a specific order. Term: each number in a sequence Arithmetic Sequence: a sequence in which each term after the first term is found by adding a constant, called the common difference (d), to the previous term.

Arithmetic Sequences (cont.) 150, 300, 450, 600, 750 … a 1 First term a 2 Second term a3a3 a4a4 a n represents a general term (nth term) where n can be any number. anan a5a5 …

3, 7, 11, 15, 19 … Notice in this sequence that if we find the difference between any term and the term before it we always get 4. 4 is then called the common difference and is denoted with the letter d. d = 4 To get to the next term in the sequence we would add 4 so a recursive formula for this sequence is: The first term in the sequence would be a 1 which is sometimes just written as a. a = 3

3, 7, 11, 15, 19 … +4 Each time you want another term in the sequence you’d add d. This would mean the second term was the first term plus d. The third term is the first term plus d plus d (added twice). The fourth term is the first term plus d plus d plus d (added three times). So you can see to get the nth term we’d take the first term and add d (n - 1) times. d = 4 Try this to get the 5th term. a 1 = 3

Let’s look at a formula for an arithmetic sequence and see what it tells us. Substituting in the set of positive integers we get: 3, 7, 11, 15, 19, … What is the common difference? d = 4 you can see what the common difference will be in the formula We can think of this as a “compensating term”. Without it the sequence would start at 4 but this gets it started where we want it. 4n would generate the multiples of 4. With the - 1 on the end, everything is back one. What would you do if you wanted the sequence 2, 6, 10, 14, 18,...?

One term of an arithmetic sequence is a 19 = 48. The common difference is d = 3. a n = a 1 + (n – 1)d a 19 = a 1 + (19 – 1)d 48 = a (3) Write general rule. Substitute 19 for n Solve for a 1. So, a rule for the n th term is: a. Write a rule for the nth term. b. Graph the sequence. –6 = a 1 Substitute 48 for a 19 and 3 for d. SOLUTION a. Use the general rule to find the first term. Example

a n = a 1 + (n – 1)d = –6 + (n – 1)3 = –9 + 3n Write general rule. Substitute –6 for a 1 and 3 for d. Simplify. Create a table of values for the sequence. The graph of the first 6 terms of the sequence is shown. Notice that the points lie on a line. This is true for any arithmetic sequence. b. Example

Recall that linear functions have a constant first difference. Notice also that when you graph the ordered pairs (n, a n ) of an arithmetic sequence, the points lie on a straight line. Thus, you can think of an arithmetic sequence as a linear function with sequential natural numbers as the domain.

Find the nth term from a Graph Example Find a formula for the nth term of the sequence graphed below.

Find the nth term from a Graph Solution The equation of the dashed line shown Below is y = –.5x +4. The sequence is given by a n = –.5n +4 for n = 1, 2, 3, 4, 5, 6.

Arithmetic Series Arithmetic Sequence Arithmetic Series. 5, 8, 11, 14, –9, –3, 3 –9 + (–3) + 3 S n represents the sum of the first n terms of a series. For example, S 4 is the sum of the first four terms. An arithmetic sequence, S n has n terms and the sum of the first and last terms is a 1 + a n.

Often in applications we will want the sum of a certain number of terms in an arithmetic sequence. The story is told of a grade school teacher In the 1700's that wanted to keep her class busy while she graded papers so she asked them to add up all of the numbers from 1 to 100. These numbers are an arithmetic sequence with common difference 1. Carl Friedrich Gauss was in the class and had the answer in a minute or two (remember no calculators in those days). This is what he did: sum is 101 With 100 numbers there are 50 pairs that add up to (101) = 5050

This will always work with an arithmetic sequence. The formula for the sum of n terms is: n is the number of terms so n/2 would be the number of pairs first term last term Let’s find the sum of But how many terms are there? We can write a formula for the sequence and then figure out what term number 59 is.

first term last term Let’s find the sum of The common difference is 2 and the first term is one so: Set this equal to 59 to find n. Remember n is the term number. 2n - 1 = 59 n = 30 So there are 30 terms to sum up.

Now Let’s Practice

Let’s try something a little trickier. What if we just know a couple of terms and they aren’t consecutive? The fourth term is 3 and the 20th term is 35. Find the first term and both a term generating formula and a recursive formula for this sequence. How many differences would you add to get from the 4th term to the 20th term? 353 Solve this for d d = 2 The fourth term is the first term plus 3 common differences. 3 (2) We have all the info we need to express these sequences. We’ll do it on next slide.

The fourth term is 3 and the 20th term is 35. Find the first term and both a term generating formula and a recursive formula for this sequence. d = 2 makes the common difference 2 makes the first term - 3 instead of 2 The recursive formula would be: Let’s check it out. If we find n = 4 we should get the 4th term and n = 20 should generate the 20th term.

The center section of a concert hall has 15 seats in the first row and 2 additional seats in each subsequent row. Theater Application How many seats are in the 20th row? Write a general rule using a 1 = 15 and d = 2. a n = a 1 + (n – 1)d a 20 = 15 + (20 – 1)(2) = = 53 Explicit rule for nth term Substitute. Simplify. There are 53 seats in the 20th row.

Theater Application How many seats in total are in the first 20 rows? Find S 20 using the formula for finding the sum of the first n terms. There are 680 seats in rows 1 through 20. Formula for first n terms Substitute. Simplify.

Check It Out! What if...? The number of seats in the first row of a theater has 14 seats. Suppose that each row after the first had 2 additional seats. How many seats would be in the 14th row? Write a general rule using a 1 = 14 and d = 2. a n = a 1 + (n – 1)d a 14 = 11 + (14 – 1)(2) = = 37 Explicit rule for nth term Substitute. Simplify. There are 37 seats in the 14th row.

Check It Out! Example 6b How many seats in total are in the first 14 rows? Find S 14 using the formula for finding the sum of the first n terms. There are 336 total seats in rows 1 through 14. Formula for first n terms Substitute. Simplify.

5. How many seats are in the theater? A movie theater has 24 seats in the first row and each successive row contains one additional seat. There are 30 rows in all. ANSWER 1155 seats 4. Write a rule for the number of seats in the nth row. ANSWER a n = 23 + n

Arithmetic Series The number of seats in the rows of the amphitheater form an arithmetic sequence. To find the number of people who could sit in the first four rows, add the first four terms of the sequence. That sum is or 96. The number of seats in the rows of the amphitheater form an arithmetic sequence. To find the number of people who could sit in the first four rows, add the first four terms of the sequence. That sum is or 96. A series is an indicated sums of the terms of a sequence. A series is an indicated sums of the terms of a sequence. Since 18, 22, 26, 30 is an arithmetic sequence, is an arithmetic series. Since 18, 22, 26, 30 is an arithmetic sequence, is an arithmetic series.

Arithmetic Series = 28 Series: the sum of the terms in a sequence. Arithmetic Series: the sum of the terms in an arithmetic sequence.

Arithmetic Series Arithmetic sequence: 2, 4, 6, 8, 10 Corresponding arith. series: Arith. Sequence: -8, -3, 2, 7 Arith. Series:

Arithmetic Series S n is the symbol used to represent the first ‘n’ terms of a series. Given the sequence 1, 11, 21, 31, 41, 51, 61, 71, … find S 4 We add the first four terms = 64

Arithmetic Series Find S 8 of the arithmetic sequence 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … = 36

Arithmetic Series The African-American celebration of Kwanzaa involves the lighting of candles every night for seven nights. The first night one candle is lit and blown out.

Arithmetic Series The second night a new candle and the candle from the first night are lit and blown out. The third night a new candle and the two candles from the second night are lit and blown out.

Arithmetic Series This process continues for the seven nights. We want to know the total number of lightings during the seven nights of celebration.

Arithmetic Series The first night one candle was lit, the 2nd night two candles were lit, the 3rd night 3 candles were lit, etc. So to find the total number of lightings we would add: