Three Consecutive Numbers

The three consecutive numbers in different patterns are explained here.

(i) The sum of any three consecutive numbers is three times of the middle number:

1 + 2 + 3 = 6 = 3 x 2

2 + 3 + 4 = 9 = 3 x 3

7 + 8 + 9 = 24 = 3 x 8

24 + 25 + 26 = 75 = 3 x 25

37 + 38 + 39 = 114 = 3 x 38

(ii) The sum of any four consecutive numbers is twice the sum of two middle numbers:

1 + 2 + 3 + 4 = 10 = 2 (2 + 3) = 2 x 5

2 + 3 + 4 + 5 = 14 = 2 (3 + 4) = 2 x 7

5 + 6 + 7 + 8 = 26 = 2 (6 + 7) = 2 x 13

17 + 18 + 19 + 20 = 74 = 2 (18 + 19) = 2 x 37

40 + 41 + 42 + 43 = 166 = 2 (41 + 42) = 2 x 83

(iii) The sum of any five consecutive numbers is five times the exact middle number:

1 + 2 + 3 + 4 + 5 = 15 = 5 x 3 = (5 x 6)/2

2 + 3 + 4 + 5 + 6 = 20 = 5 x 4 = (6 x 7)/2 – 1

3 + 4 + 5 + 6 + 7 = 25 = 5 x 5 = (7 x 8)/2 - (1 + 2)

18 + 19 + 20 + 21 + 22 = 100 = 5 x 20 = (22 x 23)/2 – (17 x 18)/2

33 + 34 + 35 + 36 + 37 = 175 = 5 x 35

110 + 111 + 112 + 113 + 114 = 560 = 5 x 112

(iv) The same pattern may be seen in the sum of six, seven, eight, ……, etc, consecutive numbers.