Warm up Determine the rule, and find the next four terms of the sequence 17, 12, 7, … -129, -98, -67,… 244, 187, 130, …

7.2 Arithmetic Sequences & Series

Arithmetic Sequences A sequence in which the difference between successive terms is a constant is called an arithmetic sequence The constant is referred to as the common difference, denoted d. To find the common difference of an arithmetic sequence, subtract any term form its succeeding term. To find the next term in the sequence, add the given term.

How to find the nth term We know that 𝑎 𝑛 = 𝑎 𝑛−1 +𝑑 Let’s consider an arithmetic sequence in which 𝑎 1 =6 and 𝑑=3 List the first five terms What pattern do you see?

Explicit Recursive Formula Find both an explicit formula and a recursive formula for the nth term of the arithmetic sequence” 12, 21, 30,… 35, 23, 11,…

Find the 68th term of the arithmetic sequence 25, 17, 9, … Find the first term of the arithmetic sequence for which 𝑎 25 =139, and 𝑑= 3 4 Find the 38th term of the arithmetic sequence -29,-2 , 25, … Find d of the arithmetic sequence for which 𝑎 1 =75 and 𝑎 38 =56.5

Arithmetic means If two nonconsecutive terms of an arithmetic sequence are known, the terms between them can be calculated. These terms are called arithmetic means. Write an arithmetic sequence that has four arithmetic means between 4.3 and 12.8 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑

Write a sequence that has six arithmetic means between 12.4 and -24.7

Arithmetic Series An arithmetic series is an indicated sum of the terms of an arithmetic sequence.

Sum of arithmetic series Find the indicated sum of each arithmetic series −5+2+9+…+317 The 28th partial sum of 27+14+1+… 𝑛=6 28 (5𝑛−17) 211 + 193 + 175 + … + (-455) The 19th partial sum of -19 + 23 + 65 +… 𝑛=23 37 (2𝑛+3) 𝑛=12 18 (−2𝑛+57)

Real world example A video gam tournament, in which gamers compete in multiple games and accumulate an overall score, pays the top 20 finishers. First place receives $5000, second place receives $4600, and so on. How much total prize money is awarded?

Real world problem Selma is playing a video game. She scores 50 points if she clears the first level. Each following level is worth 50 more points than the previous level. Thus, she scores 100 points for clearing the second level, 150 for clearing the third level, and so on. What is the total amount of points Selma will score after she clears the ninth level?

Real world problem Carter has been collecting baseball cards since his father gave him a 20-card collection. During each month, Carter’s father gives him 5 more cards than the previous month. In how many months will carter reach 1000 cards?

Real world problem Kevin runs a law mowing service. He currently has 14 clients. He has gained 2 new clients at the beginning of each of the past three years. Each year, he mows each client’s lawn an average of 15 times. Starting now, if Kevin continues to gain 2 clients each year and if he charges $30 per lawn, after how many years will he earn a total of $51,300?