This document contains a math lesson on summations. It provides formulas for calculating the sums of sequential integers from 1 to n, odd integers from 1 to 2n-1, and even integers from 2 to 2n. Several practice problems are worked through applying these formulas to calculate sums such as S = 1 + 2 + 3 + ... + 57 and S = 2 + 4 + 6 + ... + 222. The solutions are provided after each problem.

1. 2º DE SECUNDARIA
ÁREA : MATEMÁTICA
SEMANA 5
EXPERIENCIA 2
ACTIVIDAD 4
SUMATORIAS
DEMETRIO CCESA RAYME

2. 𝟏 + 𝟐 + 𝟑 + ⋯ + 𝒏 =
𝒏(𝒏 + 𝟏)
𝟐

3. S = 1 + 2 + 3 + ............ + 57
A) 1655 B) 1654 C) 1653 D) 1652

4. S = 1 + 2 + 3 + ............ + 57
A) 1655 B) 1654 C) 1653 D) 1652
S =
𝟓𝟕(𝟓𝟕+𝟏)
𝟐
=
𝟓𝟕(𝟓𝟖)
𝟐
=1653
Solución

5. S = 1 + 2 + 3 + ............ + 89
A) 4005 B) 4050 C) 4500 D) 4675

6. S = 1 + 2 + 3 + ............ + 89
A) 4005 B) 4050 C) 4500 D) 4675
S =
𝟖𝟗(𝟖𝟗+𝟏)
𝟐
=
𝟖𝟗(𝟗𝟎)
𝟐
=4005
Solución

7. S = 1 + 2 + 3 + ............ + 100
A) 5500 B) 5550 C) 5050 D) 5150

8. S = 1 + 2 + 3 + ............ + 100
A) 5500 B) 5550 C) 5050 D) 5150
S =
𝟏𝟎𝟎(𝟏𝟎𝟎+𝟏)
𝟐
=
𝟏𝟎𝟎(𝟏𝟎𝟏)
𝟐
=5050
Solución

9. S = 1 + 2 + 3 + ............ + 666
A) 222444 B) 222333 C) 222222 D) 222111

10. S = 1 + 2 + 3 + ............ + 666
A) 222444 B) 222333 C) 222222 D) 222111
S =
𝟔𝟔𝟔(𝟔𝟔𝟔+𝟏)
𝟐
=
𝟔𝟔𝟔(𝟔𝟔𝟕)
𝟐
=222111
Solución

11. S = 20 + 21 + 22 + ............ + 60
A) 1520 B) 1590 C) 1640 D) 1710

12. S = 20 + 21 + 22 + ............ + 60
A) 1520 B) 1590 C) 1640 D) 1710
S =
𝟔𝟎(𝟔𝟎+𝟏)
𝟐
=
𝟔𝟎(𝟔𝟏)
𝟐
=1830
S =
𝟏𝟗(𝟏𝟗+𝟏)
𝟐
=
𝟏𝟗(𝟐𝟎)
𝟐
=190
S = 1830 – 190 = 1640
Solución

13. S = 37 + 38 + 39 + ............ + 234
A) 26729 B) 26829 C) 26929 D) 26999

14. S = 37 + 38 + 39 + ............ + 234
A) 26729 B) 26829 C) 26929 D) 26999
S =
𝟐𝟑𝟒(𝟐𝟑𝟒+𝟏)
𝟐
=
𝟐𝟑𝟒(𝟐𝟑𝟓)
𝟐
=27495
S =
𝟑𝟔(𝟑𝟔+𝟏)
𝟐
=
𝟑𝟔(𝟑𝟕)
𝟐
=666
S = 27495 – 666 = 26829
Solución

15. 𝟐 + 𝟒 + 𝟔 + 𝟖 + ⋯ + 𝟐𝒏 = 𝒏(𝒏 + 𝟏)

16. S = 2 + 4 + 6 + ............ + 64
A) 1506 B) 1056 C) 1605 D) 1606

17. S = 2 + 4 + 6 + ............ + 64
A) 1506 B) 1056 C) 1605 D) 1606
S = 𝟑𝟐(𝟑𝟐 + 𝟏)= 32(𝟑𝟑)=1056
2n = 64
n = 32
Solución

18. S = 2 + 4 + 6 + ............ + 222
A) 12412 B) 12422 C) 12432 D) 12442

19. S = 2 + 4 + 6 + ............ + 222
A) 12412 B) 12422 C) 12432 D) 12442
S = 𝟏𝟏𝟏(𝟏𝟏𝟏 + 𝟏)= 111(𝟏𝟏𝟐)=12432
2n = 222
n = 111
Solución

20. S = 2 + 4 + 6 + ............ + 432
A) 46872 B) 46772 C) 46672 D) 46572

21. S = 2 + 4 + 6 + ............ + 432
A) 46872 B) 46772 C) 46672 D) 46572
S = 𝟐𝟏𝟔(𝟐𝟏𝟔 + 𝟏)= 216(𝟐𝟏𝟕)=46872
2n = 432
n = 216
Solución

22. S = 10 + 12 + 14 + ............ + 66
A) 1021 B) 1102 C) 1122 D) 1022

23. S = 10 + 12 + 14 + ............ + 66
A) 1021 B) 1102 C) 1122 D) 1022
S = 𝟑𝟑(𝟑𝟑 + 𝟏)= 33(𝟑𝟒)=1122
2n = 66
n = 33
2n = 8
n = 4
S = 𝟒(𝟒 + 𝟏)= 4(𝟓)=20
S = 1122 – 20 = 1102
Solución

24. S = 42 + 44 + 46 + ............ + 144
A) 4096 B) 4816 C) 4826 D) 4836

25. S = 42 + 44 + 46 + ............ + 144
A) 4096 B) 4816 C) 4826 D) 4836
S = 𝟕𝟐(𝟕𝟐 + 𝟏)= 72(𝟕𝟑)=5256
2n = 144
n = 72
2n = 40
n = 20
S = 𝟐𝟎(𝟐𝟎 + 𝟏)= 20(𝟐𝟏)=420
S = 5256 – 420 = 4836
Solución

26. S = 74 + 76 + 78 + ............ + 204
A) 9104 B) 9204 C) 9304 D) 9404

27. S = 74 + 76 + 78 + ............ + 204
A) 9104 B) 9204 C) 9304 D) 9404
S = 𝟏𝟎𝟐(𝟏𝟎𝟐 + 𝟏)= 102(𝟏𝟎𝟑)=10506
2n = 204
n = 102
2n = 72
n = 36
S = 𝟑𝟔(𝟑𝟔 + 𝟏)= 36(𝟑𝟕)=1332
S = 10506 – 1332 = 9404
Solución

28. 𝟏 + 𝟑 + 𝟓 + 𝟕 + ⋯ + 𝟐𝒏 − 𝟏 = 𝒏𝟐

29. S = 1 + 3 + 5 + ............ + 51
A) 756 B) 765 C) 676 D) 729

30. S = 1 + 3 + 5 + ............ + 51
A) 756 B) 765 C) 676 D) 729
2n – 1 = 51
2n = 52
n = 26
𝐒 = (𝟐𝟔)𝟐
= 𝟔𝟕𝟔
Solución

31. S = 1 + 3 + 5 + ............ + 127
A) 3600 B) 3906 C) 3969 D) 4096

32. S = 1 + 3 + 5 + ............ + 127
A) 3600 B) 3906 C) 3969 D) 4096
2n – 1 = 127
2n = 128
n = 64
𝐒 = (𝟔𝟒)𝟐
= 𝟒𝟎𝟗𝟔
Solución

33. S = 25 + 27 + 29 + ............ + 109
A) 2881 B) 2882 C) 2883 D) 2884

34. S = 25 + 27 + 29 + ............ + 109
A) 2881 B) 2882 C) 2883 D) 2884
2n – 1 = 109
2n = 110
n = 55
𝐒 = (𝟓𝟓)𝟐 = 𝟑𝟎𝟐𝟓
Solución
2n – 1 = 23
2n = 24
n = 12
𝐒 = (𝟏𝟐)𝟐 = 𝟏𝟒𝟒
S = 3025 – 144 = 2881

35. S = 51 + 52 + 53 + ............ + 161
A) 5926 B) 5936 C) 5946 D) 5956

36. S = 51 + 52 + 53 + ............ + 161
A) 5926 B) 5936 C) 5946 D) 5956
2n – 1 = 161
2n = 162
n = 81
𝐒 = (𝟖𝟏)𝟐 = 𝟔𝟓𝟔𝟏
Solución
2n – 1 = 49
2n = 50
n = 25
𝐒 = (𝟐𝟓)𝟐 = 𝟔𝟐𝟓
S = 6561 – 625 = 5936