Grade 10 Mathematics Arithmetic Series Lesson

An arithmetic series is the sum of a sequence in which each term is computed from the previous one by adding (or subtracting) a constant.

Arithmetic Sequence Formulas
nth Term Formula an = a1 + (n – 1)d

Sum of First n Terms Sn = n/2 (first term + last term)

1. ARITHMETIC SERIES

2. There was once a schoolboy who was asked by his teacher to give the sum of the first 100 counting numbers. 1 + 2 + 3 + … + 98 + 99 + 100

3. 1 + 2 + 3 +…+ 50 + 51 +… + 98 + 99 + 100 101 101 101 101 101 _x 50 pairings_ 5,050

4. Johann Carl Friedrich Gauss

5. Arithmetic SEries It is the sum of an arithmetic sequence

6. Arithmetic Series A series such as (3 + 7 + 11 + 15 + ··· + 99 or 10 + 20 + 30 + ··· + 1000) which has a constant difference between terms. first term is a 1 common difference is d number of terms is n last term is a n sum of an arithmetic series is S n

7. Arithmetic SEries The  sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.

8. Arithmetic SERIES S n = (a 1 + a n )   Formula ( FINITE SEQUENCE) ~ when a n is given

9. Arithmetic SERIES Formula (INFINITE SEQUENCE) S n = [2a 1 + (n - 1) d]  

10. Arithmetic Series 3 + 8 + 13 + … + 73 a 1 = 3 a n = 73 d = 5 n = ? S n = ? Example 1

11. Arithmetic SERIES S n = (a 1 + a n ) S 15 = (3 + 73) S 15 = (76) S 15 = (38) S 15 = 57   Example 1

12. Arithmetic Series 3 + 7 + 11 + 15 + ··· + 99

13. Arithmetic Series + + 1 +··· +  

14. Arithmetic Series Find the sum of the sequence -8, -5, -2, ..., 7

15. 1. integers from 1 to 50 2 . odd integers from 1 to 100 3. even integers between 1 and 101 9. 1 + 5 + 9 + … + 49 + 53 10. …  

16. 1. integers from 1 to 50 Given: a 1= 1 a n= 50 d=1 n= 50 S n = 1275 n=? S n= ?

17. 2. odd integers from 1 to 100 Given: a 1= 1 a n= 99 d=2 n= 50 S n = 2500 n=? S n= ?

18. 3. even integers between 1 and 101 Given: a 1= 2 a n= 100 d=2 n= 50 S n = 2550 n=? S n= ?

19. 9.) 1 + 5 + 9 + … + 49 + 53 Given: a 1= 1 a n= 53 d=4 n= 14 S n = 378 n=? S n= ?

20. 10. …   Given: a 1= a n= d=1   n= 10 S n = 50 n=? S n= ?

21. Arithmetic SERIES Formula (INFINITE SEQUENCE) S n = [2a 1 + (n - 1) d]  

22. Arithmetic Series What is the sum of the first 10 terms in the series: 3, 7, 11, … a 1 = 3 d = 4 n = 10 S n = ? Example

23. Arithmetic Series S n = [2a 1 + (n - 1) d] S 10 = [ 2(3) + (10 - 1) 4] S 10 = [6 + (9) 4] S 10 = [6 + 36] S 10 = [42]   Solution 1 S 10 = S 10 =  

24. Arithmetic Series What is the sum of the first 10 terms in the series: 3, 7, 11, … a 1 = 3 d = 4 n = 10 a n = ? S n = ? Solution 2

25. Arithmetic Series a n = a1 + (n – 1) d a 10 = 3 + (10 – 1) 4 a 10 = 3 + (9) 4 a 10 = 3 + 36 a 10 = 39 Solution 2

26. Arithmetic Series S n = (a 1 + a n ) S 10 = (3 + 39) S 10 = 5 (42) S 10 =   Solution 2

27. Arithmetic Series Find the sum of the first 12 positive even integers.

28. Arithmetic Series Find the sum of the first 10 negative integers

29. Arithmetic Series Find the sum of the first 20 terms of the sequence 4, 6, 8, 10, ...

30. Arithmetic Series Find the sum of the first nine terms of the arithmetic sequence with and .  

31. Arithmetic Series Find the sum of the first five terms of the arithmetic sequence with and .  

32. ARITHMETIC SERIES