4-7 Arithmetic Sequences

During a thunderstorm, you can estimate your distance from a lightning strike by counting the number of seconds from the time you see the lightning until you hear the thunder. When you list the times and distances in order, each list forms a sequence. A sequence is a list of numbers that often forms a pattern. Each number in a sequence is called a term.

Time (s) 1 2 3 4 5 6 7 8 Time (s) Distance (mi) Distance (mi) 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 +0.2 In the distance sequence, each distance is 0.2 mi greater than the previous distance. When the terms of a sequence differ by the same nonzero number d, the sequence is an arithmetic sequence and d is the common difference. The distances in the table form an arithmetic sequence with d = 0.2.

Identifying Arithmetic Sequences Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. 9, 13, 17, 21,… Step 1 Find the difference between successive terms. You add 4 to each term to find the next term. The common difference is 4. 9, 13, 17, 21,… +4 The sequence appears to be an arithmetic sequence with a common difference of 4. The next three terms are 25, 29, 33.

Reading Math The three dots at the end of a sequence are called an ellipsis. They mean that the sequence continues and can read as “and so on.”

Identifying Arithmetic Sequences Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. 10, 8, 5, 1,… Find the difference between successive terms. 10, 8, 5, 1,… The difference between successive terms is not the same. –2 –3 –4 This sequence is not an arithmetic sequence.

Example Determine whether the sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms. –4, –2, 1, 5,… Step 1 Find the difference between successive terms. –4, –2, 1, 5,… The difference between successive terms is not the same. +2 +3 +4 This sequence is not an arithmetic sequence.

Finding the nth Term of an Arithmetic Sequence Find the 8th indicated term of the arithmetic sequence. 4, 8, 12, 16, … Step 1 Find the common difference. 4, 8, 12, 16,… The common difference is 4. +4 +4 +4 The 8th term of the sequence is 32.

Example Find the indicated term of the arithmetic sequence. 7th term: 11, 5, –1, –7, … Step 1 Find the common difference. 11, 5, –1, –7,… The common difference is –6. –6 –6 –6 The 7th term of the sequence is – 26.

At the beginning of the 5th day the bag Application A bag of cat food weighs 18 pounds at the beginning of day 1. Each day, the cats are fed 0.5 pound of food. How much does the bag of cat food weigh at the beginning of day 5? Notice that the sequence for the situation is arithmetic with d = –0.5 because the amount of cat food decreases by 0.5 pound each day. At the beginning of the 5th day the bag of cat food will weigh 16 pounds.

The load will weigh 750 pounds after the 6th stop. Example Each time a truck stops, it drops off 250 pounds of cargo. After stop 1, its cargo weighed 2000 pounds. How much does the load weigh after stop 6? Notice that the sequence for the situation is arithmetic because the load decreases by 250 pounds at each stop. Since the load will be decreasing by 250 pounds at each stop, d = –250. The load will weigh 750 pounds after the 6th stop.

Lesson Review Determine whether each sequence appears to be an arithmetic sequence. If so, find the common difference and the next three terms in the sequence. not arithmetic 1) 3, 9, 27, 81,… Arithmetic; common difference is 1.5; the next three terms are 11, 12.5, 14. 2) 5, 6.5, 8, 9.5,… 3) On day 1, Zelle has knitted 60 rows of a scarf. Each day she adds 15 more rows. How many rows total has Zelle knitted on day 5? 120 rows

10-7 Geometric Sequences

The table shows the heights of a bungee jumper’s bounces. The height of the bounces shown in the table above form a geometric sequence. In a geometric sequence, the ratio of successive terms is the same number r, called the common ratio.

Geometric sequences can be thought of as functions Geometric sequences can be thought of as functions. The term number, or position in the sequence, is the input, and the term itself is the output. 1 2 3 4 Position 3 6 12 24 Term a1 a2 a3 a4 To find a term in a geometric sequence, multiply the previous term by r.

The variable a is often used to represent terms in a sequence The variable a is often used to represent terms in a sequence. The variable a4 (read “a sub 4”)is the fourth term in a sequence. Writing Math

Extending Geometric Sequences Find the next three terms in the geometric sequence. 1, 4, 16, 64,… Step 1 Find the value of r by dividing each term by the one before it. 1 4 16 64 The value of r is 4. The next three terms are 256, 1024, and 4096.

When the terms in a geometric sequence alternate between positive and negative, the value of r is negative. Helpful Hint

Extending Geometric Sequences Find the next three terms in the geometric sequence. (I want your answers as fractions!) The next three terms are

Example Find the next three terms in the geometric sequence. 5, –10, 20,–40,… The next three terms are 80, –160, and 320.

Example Find the next three terms in the geometric sequence. 512, 384, 288,… Step 2 Multiply each term by 0.75 to find the next three terms. 288 216 162 121.5 0.75 0.75 0.75 The next three terms are 216, 162, and 121.5.

Finding the nth Term of a Geometric Sequence What is the 5th term of the geometric sequence 2, –6, 18, –54, …? 2 –6 18 –54 The value of r is –3. The 9th term of the sequence is – 486.

When writing a function rule for a sequence with a negative common ratio, remember to enclose r in parentheses. –212 ≠ (–2)12 Caution

Example What is the 6th term of the sequence 1000, 500, 250, 125, …? 1000 500 250 125 The value of r is . The 6th term of the sequence is 31.25.

Application A ball is dropped from a tower. The table shows the heights of the balls bounces, which form a geometric sequence. What is the height of the 6th bounce? Bounce Height (cm) 1 300 2 150 3 75 300 150 75 The value of r is 0.5. The height of the 6th bounce is 9.375 cm.

Your Turn The table shows a car’s value for 3 years after it is purchased. The values form a geometric sequence. How much will the car be worth in the 5th year? Year Value ($) 1 10,000 2 8,000 3 6,400 10,000 8,000 6,400 The value of r is 0.8. The car will be worth $4096 in the 5th year.

Lesson Review Find the next three terms in each geometric sequence. 1. 3, 15, 75, 375,… 2. 3. The first term of a geometric sequence is 300 and the common ratio is 0.6. What is the 4th term of the sequence? 4. What is the 7th term of the sequence 4, –8, 16, –32, 64…? 1875; 9375; 46,875 64.8 256

Lesson Review 5. The table shows a car’s value for three years after it is purchased. The values form a geometric sequence. How much will the car be worth after 6 years? Year Value ($) 1 18,000 2 15,300 3 13,005 $7986.70

Assignment Study Guides 4-7 & 10-7 Skills Practice Worksheet 4-7 & 10-7