Warm-Up Find the sixth term of the sequence 𝑎 𝑛 = 1 2𝑛 + 2 Homework Answers 2) 48, −96, 192, −384 4) 21,31,43,57 8) 5, 3,−1,−9 10) 2,1,4,3 28) 20 35) 6 39) −17 42) −20

10.2: Arithmetic Sequences & Series Learning Targets Day 1: Find nth terms, common differences, and explicit formulas for arithmetic sequences and series

Arithmetic Sequence when the difference between consecutive terms is constant pattern is adding or subtracting the same value arithmetic sequences are linear function Examples: 2, 8, 14, 20, … 21, 16, 11, 6, …

Common Difference d the constant difference between terms subtract any term from its previous term To find the next term in the sequence, add the common difference to the given term.

Example 1 Determine the common difference and the next four terms of the arithmetic sequence A)–53, –36, –19, … B) 74, 68, 62, 56, …

Explicit Formula for Arithmetic Sequences 𝑎 𝑛 = 𝑎 1 + 𝑛−1 𝑑 Other ways this formula can be used: 𝑎 𝑛 = 𝑎 7 + 𝑛−7 𝑑 𝑎 19 = 𝑎 1 + 19−1 𝑑 Value of nth term Common difference Value of 1st term Term number

Example 2 Find both an explicit formula and a recursive formula for the nth term of the arithmetic sequence A) 14, 3, –8, … B) 15, 33, 51, …

Example 3 A) Find the 41st term of the arithmetic sequence 11, 4, –3, –10, …

B) Find the first term of the arithmetic sequence for which a44 = 229 and d = 8.

C) Find d of the arithmetic sequence for which a1 = 75 and a38 = 56.5 Difference Formula: *use for nonconsecutive terms 𝑑= 𝑎 𝑛 − 𝑎 𝑛 𝑛−𝑛

D) Find n if an = 336, a1 = 18, and d = 6.

E) Given a8 = 25 and a20 = 61, find a53 for the arithmetic sequence.

f) Which term of the arithmetic sequence 10, 6, 2, −2, …-146 is −146?

g) Which term of the arithmetic sequence 0. 2, 5. 2, 10. 2, 15 g) Which term of the arithmetic sequence 0.2, 5.2, 10.2, 15.2, … 225.2 is 225.2?

Exit Slip Write an explicit formula for finding the nth term of the arithmetic sequence 2, 7, 12, 17, …

10.2 Day 1 Homework Answers 4) d = 22; −17, 5, 27, 49 6) d = 17; 55, 72, 89, 106 10) 𝑎 𝑛 =3n−1 14) 𝑎 𝑛 =−14n+39 18) 𝑎 1 =−32 19) 𝑑=−7 𝑛=11 21) 𝑎 12 =−8 22) 𝑑=4

10.2: Arithmetic Sequences & Series Learning Targets Day 2: Find sums of n terms of arithmetic series

Arithmetic Series The sum of the terms in an arithmetic sequence Examples: 4.25+4+3.75+3.5 −10−12−14−16…

Sum Formula for Finite Arithmetic Series 𝑆 𝑛 = 𝑛 2 ( 𝑎 1 + 𝑎 𝑛 ) Example: Find the sum of the 1st 8 terms: −4−2+0+2+4+6+8+10

Example 6 A) Find the sum of –3 + 2 + 7 + 12 + … + 157.

b) Find the sum of 211+193+175+…−455

c) Find the 17th partial sum (S17) of the arithmetic series 53 + 31 + 9 + …

d) Find 𝑆 82 for the arithmetic series −8−8.5−9−9.5…

Exit Slip Find 𝑆 13 for –5 + 1 + 7 + …

Warm-Up 1)Evaluate: 𝑛=1 30 5𝑛+3 2) Solve using the quadratic formula: −10=−2 𝑥 2 −7𝑥 𝑥= −𝑏± 𝑏 2 −4𝑎𝑐 2𝑎 Homework Answers: 10.2 Day 2 39) 3978 40) 3317 41) – 126 42) 54 46) 480

10.2: Arithmetic Sequences & Series Learning Targets Day 3: Solve for n given a sum of an arithmetic series

Example 7 a) 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 the third, and so on. What is the total amount of points Selma will score after she clears the ninth level?

b) A tile pattern is being created where the amount of tiles increases by the same amount in each subsequent row. The seventh row has 31 tiles and the 21st row has 87 tiles. Find the total amount of tiles necessary to create a pattern that has 45 rows.

Example 8 A toy manufacturer wants to make a profit of $50,000 the first year, $62,000 the second year, $74,000 the third year, and so on. His goal is to make a profit of at least $500,000 per year selling toys. How many years will it take for the manufacturer to reach his goal?

Example 9 Consider the arithmetic series 20 + 18 + 16 + 14 + …. Write the series using summation notation. b) Find S25 c) Find n such that Sn = –760

Example 10 Consider the arithmetic series 100 + 110 + 120 + 130 + … a)Write the series using summation notation. b) Find S18 c) Find n such that Sn = 600

Exit Slip Rewrite the series using summation notation: −6−9−12−15…