1. Warm Up Complete the table… Recursive Formula Sequence
Explicit Formula
6,8,10,12…

2. Arithmetic Sequences and Partial Sums
Keeper
GSE Accelerated Pre-Calculus

3. Arithmetic Sequences
A sequence is arithmetic if there exists a number d, called the common difference, such that an+1 = an + d

4. Example For each of the following arithmetic sequences, identify
the first term, a1, and the common difference, d.
a) 6, 10, 14, 18, 22, …
b) 0, 6, 12, 18, 24, … c)

5. nth Term of an Arithmetic Sequence
The nth term of an arithmetic sequence is given by the formula:
an = a1 + (n  1)d

6. Example
Find the 11th term of the arithmetic sequence
2, 6, 10, 14, …

7. Another Example
The 3rd term of an arithmetic sequence is 5, and the 9th term is 37. Find a1 and d and construct the sequence.

8. Find the Explicit and recursive formula for the sequence
2, 5, 8, …

9. Find the Explicit and recursive formula for the sequence
−9,−16,−23, …

10. Find the Explicit and recursive formula for the sequence
25,11,−3, …

11. Find the Explicit and recursive formula for the sequence
−18,4,26, …

12. Find the specified value for the arithmetic sequence with the given characteristics
If 𝑎 14 =85 and 𝑑=9, find 𝑎 1

13. Find the specified value for the arithmetic sequence with the given characteristics
Find 𝑑 for −24, −31, −38

14. Find the specified value for the arithmetic sequence with the given characteristics
If 𝑎 𝑛 =14, 𝑎 1 =−36, and 𝑑=5 find 𝑛

15. Find the specified value for the arithmetic sequence with the given characteristics
If 𝑎 1 =47 and 𝑑=−5, find 𝑎 12

16. Find the specified value for the arithmetic sequence with the given characteristics
If 𝑎 22 =95 and 𝑎 1 =11, find 𝑑

17. Find the specified value for the arithmetic sequence with the given characteristics
If 𝑎 𝑛 =−20, 𝑎 1 =46, and 𝑑=−11, find 𝑛.

18. Find the specified value for the arithmetic sequence with the given characteristics
If 𝑎 35 =−63 and 𝑎 1 =39, find 𝑑

19. Sum of the First n Terms
The formula for the sum of the first n terms of an arithmetic sequence is given by:

20. Example Find the sum of the first 11 terms of the arithmetic sequence
16, 12, 8, 4, …

21. Another Example
Find the sum:

22. Find the indicated sum of the arithmetic series
26th partial sum of …+303

23. Find the indicated sum of the arithmetic series
−28+ −19 + −10 +…+242

24. Find the indicated sum of the arithmetic series
42nd partial sum of …

25. Find the indicated sum of the arithmetic series
54th partial sum of …

26. Find the indicated sum of the arithmetic series
− …+649

27. Find the indicated sum of the arithmetic series
…+(−562)

28. Find the indicated sum of each arithmetic series
𝑛=1 20 (3+2𝑛)

29. Find the indicated sum of each arithmetic series
𝑛=7 18 (−9𝑛−26)

30. Find the indicated sum of each arithmetic series
𝑛=7 42 (84−3𝑛)

31. Find the indicated sum of each arithmetic series
𝑛= 𝑛 2 −9

32. Find the indicated sum of each arithmetic series
𝑛=1 13 [32+4 𝑛−1 ]

33. Example
Find 𝑎 𝑛 if 𝑆 𝑛 =490, 𝑎 1 =−5, and 𝑛=100

34. Example
If 𝑆 𝑛 =51.7, 𝑛=22, 𝑎 𝑛 =−11.3, find 𝑎 1

35. Example
Find 𝑛 for −7+ −5.5 + −4 +… if 𝑆 𝑛 =−14 and 𝑎 𝑛 =3.5

36. Example
If 𝑆 26 =1456, and 𝑎 1 =−19, find 𝑑

37. Example
If 𝑆 12 =174, 𝑎 12 =39, find 𝑑

38. Write the arithmetic series in sigma notation
…+66

39. Write the arithmetic series in sigma notation
−1+0+1+…+7

40. Write the arithmetic series in sigma notation
…+61

41. Write the arithmetic series in sigma notation
− − − …+ − 3 5

42. Write the arithmetic series in sigma notation
…−2

43. Application
An orchestra consists of 8 rows of musicians. The first row has 5 musicians, the second row has 7 musicians, and the third row has 9 musicians.
a) How many musicians are in the last row?
b) What is the total number of musicians in the orchestra?

44. Application
A crew is tiling a hotel lobby with a trapezoid mosaic pattern. The shorter base of the trapezoid begins with a row of 8 tiles. Each row has two additional tiles until the 20th row. Determine the number of tiles needed to create the mosaic design. .

45. More Practice
Write a function that can be used to model the nth term of each sequence 2,5,8,11,14,17,…

46. More Practice
Write a function that can be used to model the nth term of each sequence 8,13,20,29,40,53,…

47. Challenge
Write a function that can be used to model the nth term of each sequence 2,2,4,8,14,22,…

48. Challenge
Write a function that can be used to model the nth term of each sequence 5,31,97,221,421,715,…

49. Challenge
Write a function that can be used to model the nth term of each sequence −6,−8,−6,6,34,84,…

50. Challenge
Write a function that can be used to model the nth term of each sequence 0,23,134,447,1124,2375,…