Number Sums Investigations

23+24+25+26, 11+12+13+14+15+16+17

99

49+50, 4+5+6+7+8+9+10+11+12+13+14, 7+8+9+10+11+12+13+14+15, 14+15+16+17+18+19, 32+33+34

00

8+19+20+21+22, 9+10+11+12+13+14+15+16

Table of two number sums

Number

Sum

Number

Sum

0+1

51

25+26

3

+2

53

26+27

5

2+3

55

27+28

7

3+4

57

28+29

9

4+5

59

29+30

1

5+6

61

30+31

3

6+7

63

31+32

5

7+8

65

32+33

7

8+9

67

33+34

9

9+10

69

34+35

21

0+11

71

35+36

23

1+12

73

36+37

25

2+13

75

37+38

27

3+14

77

38+39

29

4+15

79

39+40

31

5+16

81

40+41

33

6+17

83

41+42

35

7+18

85

42+43

37

8+19

87

43+4

39

9+20

89

44+45

41

20+21

91

45+46

43

21+22

93

46+47

45

22+23

95

47+48

47

23+24

97

48+49

49

24+25

99

49+50

Table of three number sums

Number

Sum

6

+2+3

9

2+3+4

2

3+4+5

5

4+5+6

8

5+6+7

21

6+7+8

24

7+8+9

27

8+9+10

30

9+10+11

33

0+11+12

36

1+12+13

39

2+13+14

42

3+14+15

45

4+15+16

48

5+16+17

51

6+17+18

54

7+18+19

57

8+19+20

60

9+20+21

63

20+21+22

66

21+22+23

69

22+23+24

72

23+24+25

75

24+25+26

78

25+26+27

81

26+27+28

84

27+28+29

87

28+29+30

90

29+30+31

93

30+31+32

96

31+32+33

99

32+33+34

Table of four number sums

Number

Sum

0

+2+3+4

4

2+3+4+5

8

3+4+5+6

22

4+5+6+7

26

5+6+7+8

30

6+7+8+9

34

7+8+9+10

38

8+9+10+11

42

9+10+11+12

46

0+11+12+13

50

1+12+13+14

54

2+13+14+15

58

3+14+15+16

62

4+15+16+17

66

5+16+17+18

70

6+17+18+19

74

7+18+19+20

78

8+19+20+21

82

9+20+21+22

86

20+21+22+23

90

21+22+23+24

94

22+23+24+25

98

23+24+25+26

Table of five number sums

Number

Sum

5

+2+3+4+5

20

2+3+4+5+6

25

3+4+5+6+7

30

4+5+6+7+8

35

5+6+7+8+9

40

6+7+8+9+10

45

7+8+9+10+11

50

8+9+10+11+12

55

9+10+11+12+13

60

0+11+12+13+14

65

1+12+13+14+15

70

2+13+14+15+16

75

3+14+15+16+17

80

4+15+16+17+18

85

5+16+17+18+19

90

6+17+18+19+20

95

7+18+19+20+21

00

8+19+20+21+22

Table of six number sums

Number

Sum

21

+2+3+4+5+6

27

2+3+4+5+6+7

33

3+4+5+6+7+8

39

4+5+6+7+8+9

45

5+6+7+8+9+10

51

6+7+8+9+10+11

57

7+8+9+10+11+12

63

8+9+10+11+12+13

69

9+10+11+12+13+14

75

0+11+12+13+14+15

81

1+12+13+14+15+16

87

2+13+14+15+16+17

93

3+14+15+16+17+18

99

4+15+16+17+18+19

Table of seven number sums

Number

Sum

28

+2+3+4+5+6+7

35

2+3+4+5+6+7+8

42

3+4+5+6+7+8+9

49

4+5+6+7+8+9+10

56

5+6+7+8+9+10+11

63

6+7+8+9+10+11+12

70

7+8+9+10+11+12+13

77

8+9+10+11+12+13+14

84

9+10+11+12+13+14+15

91

0+11+12+13+14+15+16

98

1+12+13+14+15+16+17

Table of eight number sums

Number

Sum

36

+2+3+4+5+6+7+8

44

2+3+4+5+6+7+8+9

52

3+4+5+6+7+8+9+10

60

4+5+6+7+8+9+10+11

68

5+6+7+8+9+10+11+12

76

6+7+8+9+10+11+12+13

84

7+8+9+10+11+12+13+14

92

8+9+10+11+12+13+14+15

00

9+10+11+12+13+14+15+16

Table of nine number sums

Number

Sum

45

+2+3+4+5+6+7+8+9

54

2+3+4+5+6+7+8+9+10

63

3+4+5+6+7+8+9+10+11

72

4+5+6+7+8+9+10+11+12

81

5+6+7+8+9+10+11+12+13

90

6+7+8+9+10+11+12+13+14

99

7+8+9+10+11+12+13+14+15

Table of ten number sums

Number

Sum

55

+2+3+4+5+6+7+8+9+10

65

2+3+4+5+6+7+8+9+10+11

75

3+4+5+6+7+8+9+10+11+12

85

4+5+6+7+8+9+10+11+12+13

95

5+6+7+8+9+10+11+12+13+14

Table of eleven number sums

Number

Sum

66

+2+3+4+5+6+7+8+9+10+11

77

2+3+4+5+6+7+8+9+10+11+12

88

3+4+5+6+7+8+9+10+11+12+13

99

4+5+6+7+8+9+10+11+12+13+14

Table of twelve number sums

Number

Sum

78

+2+3+4+5+6+7+8+9+10+11+12

90

2+3+4+5+6+7+8+9+10+11+12+13

Table of thirteen number sums

Number

Sum

91

+2+3+4+5+6+7+8+9+10+11+12+13

A table of numbers that cannot be made using Consecutive Sums

Number

Sum

-

2

-

4

-

8

-

6

-

32

-

64

-

Equations

The equation for two number sums is, n+n+1, which is simplified to 2n+1. E.g.

301=2n+1

-1 -1

300=2n

2 2 n=150 150+151=301

The equation for three number sums is, n+n+1+n+2. This is simplified to 3n+3. E.g.

300=3n+3

-3 -3

297=3n

3 3 n=99 99 + 100 + 101=300

The equation for four number sums is, n+n+1+n+2+n+3, which can be simplified to 4n+6. E.g.

310=4n+6

-6 -6

304=4n

4 4 n=76 76+77+78+79=310

The equation for five number sums is, n+n+1+n+2+n+3+n+4. This can be simplified to just 5n+10. E.g.

310=5n+10

-10 -10

300=5n

5 5 n=60 60+61+62+63+64=310

The equation for six number sums is, n+n+1+n+2+n+3+n+4+n+5. This can be simplified to just 6n+15. E.g.

315=6n+15

-15 -15

300=6n

6 6 n=50 50+51+52+53+54+55=315

The equation for seven number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6. This can be simplified to just 7n+21. E.g.

350=7n+21

-21 -21

329=7n

7 7 n=47 47+48+49+50+51+52+53=350

The equation for eight number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6+n+7. This can be simplified to just 8n+28. E.g.

316=8n+28

-28 -28

288=8n

8 8 n=36 36+37+38+39+40+41+42+43=316

The equation for nine number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6+n+7+n+8. This can be simplified to just 9n+36. E.g.

360=9n+36

-36 -36

324=9n

9 9 n=36 36+37+38+39+40+41+42+43+44=360

The equation for 10 number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6+n+7+n+8+n+9. This can be simplified to 10n+45. E.g.

315=10n+45

-45 -45

270=10n

10 10 n=27 27+28+29+30+31+32+33+34+35+36=315

The equation for eleven number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6+n+7+n+8+n+9+n+10. This is simplified to 11n+55. E.g.

330=11n+55

-55 -55

275=11n

11 11 n=25 25+26+27+28+29+30+31+32+33+34+35=330

The equation for twelve number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6+n+7+n+8+n+9+n+10+n+11. This can be simplified to 12n+66. E.g.

390=12n+66

-66 -66

324=12n

12 12 n=27 27+28+29+30+31+32+33+34+35+36+37+38=390

The equation for thirteen number sums is, n+n+1+n+2+n+3+n+4+n+5+n+6+n+7+n+8+n+9+n+10+n+11+n+12. This can be simplified to just 13n+78. E.g.

351=13n+78

-78 -78

273=13n

13 13 n=21 21+22+23+24+25+26+27+28+29+30+31+32+33=351

Most numbers can be written as the sum of two or more consecutive numbers, e.g. 1+2=3, 2+3=5, 3+4=7, etc. In this project I will be investigating Consecutive Sums and the patterns that they have.

First of all I will try to find out all of the consecutive sums for the numbers 1-100. To do this I will draw a table, (below), and put in the different sums for each number up to 100.

00

Write each sum in here for each number, only if there is a sum for each number.

Once I have all of the sums in a table, I will sort them out into further tables. I.e. There will a table for numbers made with two number sums - 1+2, 2+3, 3+4, etc, a table for numbers made with three number sums - 1+2+3, 2+3+4, 3+4+5, etc. Once I have filled in all of the tables, I will make one other bigger table. This table will have the numbers 1-100 on the left side, and the numbers 0,2-13 on the top. The numbers on the top represent the amount of numbers needed in a sum to make the appropriate number, e.g.

0

2

3 - 13

00

The boxes will be coloured in appropriately.

After this I will go on to form an equation for each table, so I can find out the sum for a number, i.e. an equation to find out sums for numbers made with two number sums, three number sums, etc. When I have formed all of the equations I will show how they work by taking some large numbers and working out the sums for those numbers.

With all this information I will be able to draw conclusions. I will explain my observations for each table and describe and explain any patterns I found.

There are thirteen tables and each one of them has an individual rule for them.

The numbers that cannot be made at all are, 1, 2, 4, 8, 16, 32, and 64. All of these numbers have something in common. Firstly all of the numbers, apart from one, are all even numbers, which make them divisible by two. They can all be made by multiplying the number before that number by two, e.g.

2=1x2 = 21

4=2x2 =22

8=4x2 =23

6=8x2 =24

32=16x2 =25

64=32x2=26

Numbers that can be made using two number sums are all odd numbers. They go up in two's. All of the prime numbers up to 100 are in this table.

The three number sums all give an answer that is a multiple of three.

The four number sums all give an answer that is an even number.

Numbers made using five number sums are all multiples of five.

Six number sums have answers that all go up in six's

For the seven number sums the answers all go up in multiples of seven.

In the eight number sums the answers all go up in eights. The answers are all even numbers.

Numbers made using nine number sums are all multiples of nine.

The ten number sums all give an answer that goes up in tens each time. They are all even numbers as well as being multiples of five. The numbers all end in five.

The eleven number sums all give the answer that is a multiple of eleven.

Numbers made using twelve number sums go up in twelve's each time, but are not multiples of twelve. They are all even numbers.

For the thirteen number sums the number made is always a multiple of thirteen.

For all the odd number tables, i.e. 3,5,7 etc, the numbers made are multiples of the number of that table. E.g. for five number sums the numbers go up; 15, 20, 25, 30, etc. For the even numbers the numbers just go up by the number of table, but are not multiples. E.g. for six number sums the numbers go up; 28(+6),35(+6), 42, etc.