Challenging P6 2010 SA2 Pei Chun P2 Q18

The following figures are made of ovals.
(a) Complete the table above for Fig 5. (Ans: 41)
(b) How many ovals will there be in Fig 52? (Ans: 14695) Should be 14965
(c) Which figure will have 5305 ovals? (Ans: 5305) Should be 52

F1 --> 1
F2 --> 5 = 1 + 4
F3 --> 13 = 1 + 4 + 8
F4 --> 25 = 1 + 4 + 8 + 12

F5 --> 1 + 4 + 8 + 12 + 16 = 41

a) It was 41.


Fn --> 1 + 4 x 1 + 4 x 2 + ... + 4 x (n - 1)
= 1 + 4 + 4 x 2 + ... (4 n - 4)
= 1 + 4n x (n -1)/2
= 1 + (2 n2 - 2n)
= 2 n2 - 2 n + 1

F78 --> 2 x 78 x 78 - 2 x 78 + 1 = 14965

b) It was 14965.

2n2 - 2n + 1 --> 5305
2n2 - 2n --> 5305 - 1 = 5304
n (n - 1) --> 5304/2= 2652
52 x 51 --> 2652

c) It was figure 52.