Variations?

　　　　　　　　　　　　　　　　　　　　　　　　
Question：
Ho Man is investigating the greatest number of regions possible when a number of intersecting tangents
are drawn to a circle.
(1) He first draws a circle and a tangent on a piece of paper. He finds that the piece of paper is at most
　　divided into 3 regions, as shown in Figure 3(a).
(2) Then he draws another tangent and now the paper is at most divided into 6 regions, as shown in
　　Figure 3(b).
(3) He further draws another tangent on the paper, as shown in Figure 3(c).

30(a) Complete the following table.
30(b) It is known the greatest number of regions N after drawing n tangents on the paper varies jointly as
　 　 (n + 1) and (n + c), where c is a constant.
　 (i) Using the information found in the above table, express N in terms of n.
　(ii) Can we use 36 tangents to divide the paper into 705 regions? Explain you answer.

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Solution：
30(a)
1：3
2：6
3：10
4：15 ...... ( https://s31.postimg.org/vyp7yvbkr/1234.png )


30(bi)
N ∝ (n + 1) (n + c)
N = k (n + 1) (n + c) ...... [ k ≠ 0 is a constant ]

Sub n = 1, N = 3 into N = k (n + 1) (n + c)
3 = k (1 + 1) (1 + c)
3 = 2k + 2kc ...... ①

Sub n = 2, N = 6 into N = k (n + 1) (n + c)
6 = k (2 + 1) (2 + c)
6 = 6k + 3kc ...... ②

3① - 2②：
9 - 12 = 3(2k + 2kc) - 2(6k + 3kc)
-3 = -6k
k = 1/2

Sub k = 1/2 into ①
3 = 2(1/2) + 2(1/2)c
c = 3 - 1 = 2

∴ N = (n + 1)(n + 2)/2

30(bii)
When n = 36,
N = (36 + 1)(36 + 2)/2 = 703 < 705
∴ No

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