Lesson 5 Arithmetic Sequences.pdf
- 2. What’s on the Agenda • Vocabulary • Identifying arithmetic sequences • Finding the next term of an arithmetic sequence • Finding the nth term of an arithmetic sequence • Arithmetic sequences and functions
- 3. Vocabulary • A sequence is a set of numbers. • The terms of a sequence are the specific order of the numbers in a set. • An arithmetic sequence is a set of numbers where the difference between two successive terms is constant. • The difference between the two successive terms is known as the common difference denoted d.
- 4. Identifying an Arithmetic Sequence • Is the sequence of numbers an arithmetic sequence? 0, 2, 4, 6, 8, 10, … • The number 0 is the first term, 2 is the second term, 4 is the third term etc. • The difference between the 1st term and 2nd term, 2 – 0 = 2 • Because the difference between any two successive terms is 2, the sequence has a common difference d = 2. So the sequence is an arithmetic sequence!
- 5. Identifying an Arithmetic Sequence • Determine if the sequence is an arithmetic sequence. − 1 2 , − 3 8 , − 1 8 , 0, 1 4 , … • The difference between the first and second term is 1 8 . The difference between the second and third term is 2 8 𝑜𝑟 1 4 . Since the two differences are not equal this is not an arithmetic sequence.
- 6. Identifying an Arithmetic Sequence • Determine if the sequence is an arithmetic sequence and explain why. -20, -16, -12, -8, -4,…
- 7. Identifying an Arithmetic Sequence • Determine if the sequence is an arithmetic sequence and explain why. -20, -16, -12, -8, -4,… • The difference between any two terms is 4. • So the common difference d=4 • Since the difference between the terms in the sequence is constant this is an arithmetic sequence.
- 8. Finding the Next Term • If a sequence of numbers has a common difference d we can add d to the previous term to get the next term. • What are the 6th and 7th terms of the arithmetic sequence, -13, -8, -3, 2, 7, … • Step 1) Find the common difference d • Step 2) Add d to the 5th term to get the 6th term • Step 3) Add d to the 6th term to get the 7th term
- 9. Finding the Next Term • What are the 6th and 7th terms of the arithmetic sequence, -13, -8, -3, 2, 7, … • Step 1) Find the common difference d • d=5 • Step 2) Add d to the 5th term to get the 6th term • The 5th term is 7 so 7+5=12 • 6th term is 12 • Step 3) Add d to the 6th term to get the 7th term • 12+5=17 • 7th term is 17
- 10. Find the Next Term • What are the next three terms in the arithmetic sequence? -8.5, -9, -9.5, -10,…
- 11. Find the Next Term • What are the next three terms in the arithmetic sequence? -8.5, -9, -9.5, -10,… • d=-0.5 • The 5th term is -10.5 • The 6th term is -11 • The 7th term is -11.5
- 12. Finding the nth Term • So far we have figured out how to find next term from the sum of the previous term and the common difference. • We can actually find what any term is from the first term and the common difference! • To find the nth term of any arithmetic sequence with the first term 𝑎1and the common difference d we can use the function 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 where n is a positive integer
- 13. Where does this equation come from!? Term Symbol In Terms of 𝒂𝟏 and d In Terms of Successive Terms and d 1st term 𝒂𝟏 𝒂𝟏 𝒂𝟏 2nd term 𝒂𝟐 𝒂𝟏 + 𝒅 𝒂𝟏 + 𝒅 3rd term 𝒂𝟑 𝒂𝟏 + 𝟐𝒅 (𝒂𝟏+𝒅) + 𝒅 =𝒂𝟐 + 𝒅 4th term 𝒂𝟒 𝒂𝟏 + 𝟑𝒅 (𝒂𝟐+𝒅) + 𝒅 =𝒂𝟑 + 𝒅 ⋮ ⋮ ⋮ ⋮ nth term 𝒂𝒏 𝒂𝟏 + (𝒏 − 𝟏)𝒅 (𝒂(𝒏−𝟐)+𝒅) + 𝒅 =𝒂(𝒏−𝟏) + 𝒅
- 14. Finding the nth Term • Write an equation for the nth term of the arithmetic sequence -12, -8, -4, 0, … • Step 1) Find the common difference • 4 • Step 2) Write an equation • 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 • 𝒂𝒏 = −𝟏𝟐 + 𝒏 − 𝟏 𝟒 • 𝒂𝒏 = −𝟏𝟐 + 𝟒𝒏 − 𝟒 • 𝒂𝒏 = 𝟒𝒏 − 𝟏𝟔
- 15. Finding the nth Term • Find the 9th term of the sequence. -12, -8, -4, 0, … • Substitute 9 for n in the formula • 𝒂𝒏 = 𝟒𝒏 − 𝟏𝟔 • 𝒂𝟗 = 𝟒(𝟗) − 𝟏𝟔 • 𝒂𝟗 = 𝟐𝟎
- 16. Finding the nth Term • Graph the first five terms of the sequence. -12, -8, -4, 0, … -14 -12 -10 -8 -6 -4 -2 0 2 4 6 0 1 2 3 4 5 6 n 4n-16 𝒂𝒏 (n, 𝒂𝒏) 1 4(1)-16 -12 (1,-12) 2 4(2)-16 -8 (2,-8) 3 4(3)-16 -4 (3,-4) 4 4(4)-16 0 (4,0) 5 4(5)-16 4 (5,4)
- 17. How to Graph on Desmos (n, 𝒂𝒏) (1,-12) (2,-8) (3,-4) (4,0) (5,4)
- 18. Finding the nth Term • What term of the sequence is 32? 𝒂𝒏 = 𝟒𝒏 − 𝟏𝟔 32 = 𝟒(𝒏) − 𝟏𝟔 𝟑𝟐 + 𝟏𝟔 = 𝟒 𝒏 − 𝟏𝟔 + 𝟏𝟔 𝟒𝟖 = 𝟒 𝒏 𝟏𝟐 = 𝒏 • 32 is the 12th term of the sequence.
- 19. Finding the nth Term • Consider the arithmetic sequence 3, -10, -23, -36,… • 1) Write an equation for the nth term of the sequence • 2) Find the 15th term in the sequence. • 3) Graph the first 5 terms of the sequence. • 4) Which term of the sequence is -114?
- 20. Finding the nth Term • Consider the arithmetic sequence 3, -10, -23, -36,… • 1) Write an equation for the nth term of the sequence • d=-13 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 𝒂𝒏 = 𝟑 + 𝒏 − 𝟏 (−𝟏𝟑) 𝒂𝒏 = −𝟏𝟑𝒏 + 𝟏𝟔
- 21. Finding the nth Term • Consider the arithmetic sequence 3, -10, -23, -36,… • 2) Find the 15th term in the sequence. 𝒂𝒏 = −𝟏𝟑𝒏 + 𝟏𝟔 𝒂𝟏𝟓 = −𝟏𝟑(𝟏𝟓) + 𝟏𝟔 𝒂𝟏𝟓 = −𝟏𝟗𝟓 + 𝟏𝟔 𝒂𝟏𝟓 = −𝟏𝟕𝟗 • The 15th term of the sequence is -179.
- 22. Finding the nth Term • Consider the arithmetic sequence 3, -10, -23, -36,… • 3) Graph the first 5 terms of the sequence. n −𝟏𝟑𝒏 + 𝟏𝟔 𝒂𝒏 (n, 𝒂𝒏) 1 -13(1)+16 3 (1,3) 2 -13(2)+16 -10 (2,-10) 3 -13(3)+16 -23 (3,-23) 4 -13(4)+16 -36 (4,-36) 5 -13(5)+16 -49 (5,-49)
- 24. Finding the nth Term • Consider the arithmetic sequence 3, -10, -23, -36,… • 4) Which term of the sequence is -114? −𝟏𝟏𝟒 = −𝟏𝟑𝒏 + 𝟏𝟔 −𝟏𝟏𝟒 − 𝟏𝟔 = −𝟏𝟑𝒏 + 𝟏𝟔 − 𝟏𝟔 −𝟏𝟑𝟎 = −𝟏𝟑𝒏 1𝟎 = 𝒏 • -114 is the 10th term in the sequence.
- 25. Arithmetic Sequences and Functions • From the graph of an arithmetic sequence we see that arithmetic sequences are linear functions. • n is the x-value or independent variable • 𝒂𝒏 is the y-value or dependent variable • d the common difference is the slope. • 𝒂𝒏 = 𝒂𝟏 + 𝒏 − 𝟏 𝒅 as a linear functions is f(n)= 𝒏 − 𝟏 𝒅 + 𝒂𝟏
- 26. Arithmetic Sequences as Functions • Marisol is mailing invitations to her quinceañera. The arithmetic sequence $0.42, $0.84, $1.26, $1.68,… represents the cost of postage. • 1) Write a function to represent this sequence. • 2) Graph the function
- 27. Arithmetic Sequences as Functions • Marisol is mailing invitations to her quinceañera. The arithmetic sequence $0.42, $0.84, $1.26, $1.68,… represents the cost of postage. • 1) Write a function to represent this sequence. • f(n)= 𝒏 − 𝟏 𝒅 + 𝒂𝟏 • f(n)= 𝒏 − 𝟏 𝟎. 𝟒𝟐 + 𝟎. 𝟒𝟐 • f(n)= 𝟎. 𝟒𝟐𝒏
- 28. Arithmetic Sequences as Functions • Marisol is mailing invitations to her quinceañera. The arithmetic sequence $0.42, $0.84, $1.26, $1.68,… represents the cost of postage. • 2) Graph the function n 0. 𝟒𝟐𝒏 f(n) (n, f(n)) 1 0.42(1) 0.42 (1,0.42) 2 0.42(2) 0.84 (2,0.84) 3 0.42(3) 1.26 (3,1.26) 4 0.42(4) 1.68 (4,1.68) 5 0.42(5) 2.10 (5,1.90)
- 29. Is this arithmetic sequence a direct variation?
- 30. Review • All arithmetic sequences have a ________ which is denoted ___. • The common difference can be either a positive or negative number. (true or false) • Arithmetic sequences are always direct variations. (true or false) • You can find any term of an arithmetic sequence by ______ the common difference to the previous term. • The only things we need to determine the nth term of a sequence are the _____ term denoted 𝑎1 and the common difference.
- 31. THE BIG PICTURE LINEAR FUNCTIONS y=mx+b ax+by=c Direct Variation y=kx and contains (0,0) Arithmetic Sequences 𝑎𝑛 = 𝑛 − 1 𝑑 + 𝑎1 where n is positive