Challenging P6

6,15,27,42,.... --> 6, (6+9), (6+9+12), (6+9+12+15),.... What's the formula?


1 --> 6 = 3 + 3 x 1
            = 3 x 1 + 3 x 1
2 --> 15 = 6 + 9
              = (3 + 3 x1) + (3 + 3 x 2)
              = 3 x 2 + 3 x 1 + 3 x 2
              = 3 x 2 + 3 x (1 + 2)  

3 --> 27 = 15 + 12
              = (3 + 3 x 1) + (3 + 3 x 2) + (3 + 3 x 3)
              = 3 x 3 + 3 x 1 + 3 x 2 + 3 x 3
              = 3 x 3 + 3 x (1 + 2 + 3)

N --> 3 x N + 3 x ( 1 + 2 + ...... + N)     
       = 3 x N + 3 x (1 + N)/2 x N  You may stop here as the next step
       = 3 x N x (3 + N)/2               involves factorisation (Sec 1 level)


Testing
-------
1 --> 3 x 1 x (3 + 1)/2 = 3 x 2
                                    = 6 (correct)

2 --> 3 x 2 x (3 + 2)/2 = 15  (correct)

3 --> 3 x 3 x (3 + 3)/2 = 27 (correct)

4 --> 3 x 4 x (3 + 4)/2 = 42 (correct)

5 --> 3 x 5 x (3 + 5)/2 = 60

6 --> 3 x 6 x (3 + 6)/2 = 81

...

10 --> 3 x 10 + (3 + 10)/2 = 195


1st Fig is 1 hexagon - made up of 6 sticks
(Hexagon resting on one edge)

2nd Fig is 3 hexagon - made up of 15 sticks
(1 hexagon resting on two hexagons)

3rd Fig is 6 hexagon - made up of 27 sticks
(The two hexagons resting on 3 hexagons)

Based on figures given,

1 --> 6

2 --> 6 x (1 + 2) - 3 x 1 = 18 - 5
                                      = 5

3 --> 6 x (1 + 2 + 3) - 3 x (1 + 2) = 36 - 9
                                                     = 27

N --> 6 x (1 + 2 + ...+ N) - 3 x (1 + 2 + ...N -1)
       = 6 x (1 + N)/2 x N - 3 x (1 + N - 1)/2 x (N -1)
       = 3 x N x (N + 1) - 3 x N x (N - 1)/2
       if factorised will get the same formula as above

Testing

1 --> 3 x 1 x 2 - 3 x 1 x 0/2 = 6 (correct)

2 --> 3 x 2 x 3 - 3 x 2 x 1/2 = 15 (correct)

3 --> 3 x 3 x 4 - 3 x 3 x 2/2 = 27 (correct)

10 --> 3 x 10 x 11 - 3 x 10 x 9/2 = 195 (same as above)