The sum of the arithmetic sequence

1. Arithmetic Series- the sum of the
arithmetic sequence

2. FORMULA
General Formula
an=a1+(n-1)d
Sn = n/2 (a1 + an)
Or
Sn = n/2 [2a1 +(n-1)d

3. Example 1. 3, 11, 19
Find S6
d=8

4. an=a1+(n-1)d
a6 = 3+(6-1)8
3+(5)8
3+40
a6 = 43

5. Sn = n/2 (a1 + an)
S6 = 6/2 (3 + 43)
S6 = 3 (46)
S6 =138

6. Example #2: a1=7 and d=9
Find S9
an=a1+(n-1)d
a9 = 7+(9-1)9
7+(8)9
a9 = 72
Sn = n/2 (a1 + an)
S9 = 9/2 (7 + 79)
9(43)
S9 = 387

7. Example #3:
If you were given P50.00 on the first day of
school, P55.00 on the second day and
additional P5.00 daily, how much money
will you be able to save by the end of the
week?

8. a14= 2weeks
 an=a1+(n-1)d
 a14 = 50+(14-1)5
 50+(13)5
 50+(13)5
 a14 = 115
 Sn = n/2 (a1 + an)
 S14 = 14/2 (50 + 115)
 7(165)
 S14 = 1115

9. Example #4: find S10 if:
a. a1 = 5 and an= 68
B. a1 = 5, d= -7 an= ?
 c. 3, 5, 7, 11
D. 6, 3, 0, -3

10. a. a1 = 5 and an= 68
Sn = n/2 (a1 + an)
10/2 (5+68)
5 (73)
S10=365

11. B. a1 = 5, d= -7 n=10 an= ?
Sn = n/2 [2a1 +(n-1)d]
10/2 [2(5) +(10-1)-7]
5 [10 + 9 (-7)]
5 [ 10 + (-63) ]
5 [-53 ]
S10=-265

12. c. 3,5,7,11
a1 = 3, d= 2 n=10 an= ?
Sn = n/2 [2a1 +(n-1)d]
10/2 [2(3) +(10-1)2]
5 [6 + 9 (2)]
5 [6 + 18]
5 [24]
Sn =120

13. d. 6, 3, 0, -3
a1 = 6, d= -3 n=10 an= ?
Sn = n/2 [2a1 +(n-1)d]
10/2 [2(6) +(10-1)-3]
5 [12 + 9 (-3)]
5 [12 + (-27)]
5 [-15]
Sn =-75

14. Example #5: Find the first term and the last term
and the sum of each finite arithmetic series.
 1. 𝑛=1
31
6𝑛
 2. 𝑛=3
20
(3𝑛 − 1)
 3. 𝑛=2
16
(3𝑛 + 4)
 4

15. 1. 𝑛=1
31
6𝑛
a1 = 6n
 =6(1)
 =6
an=6n
 =6(31)
 =186
n=31

16. Sn = n/2 (a1 + an)
31/2 (6 + 186)
31/2(192)
31(96)
2976

17. 2. 𝑛=3
20
(3𝑛 − 1)
a1 = 3n-1
 =3(3)-1
 =9-1
 8
an =3n-1
 =3(20)-1
 =60-1
59
n=18 (20-3+1)=18

18. Sn = n/2 (a1 + an)
18/2 (8 + 59)
18/2(67)
9(67)
603

19. 3. 𝑛=2
16
(3𝑛 + 4)
a1 = 3n+4
 =3(2)+4
 =6+4
 =10
an=3n+4
 =3(16)+4
 =48+4
 52
n=15

20. Sn = n/2 (a1 + an)
15/2 (10 + 52)
15/2(62)
15/2(62)
15 (31)
465

21. Example number 6: Evaluate the related
series of each problem.
1. 13, 15, 17, 19, 21, 23
= 108