F.4 A-MTH M.I.

let P be the proposition "an=(5/2)(3^(n-1))-1/2 for all natural numbers n"

when n = 1
a1 = 2 = (5/2)(3^(1-1))-1/2
therefore P(1) is true

when n = 2
a2 = 3a(2-1)+1 = 7 = (5/2)(3^(2-1))-1/2
therefore P(2) is true

assume that P(k) is true,
ie. ak = (5/2)(3^(k-1))-1/2

when n = k+1
a(k+1)
= 3ak+1
= 3[(5/2)(3^(k-1))-1/2]+1 (by assumption)
= (5/2)(3^k)-3/2+1
= (5/2)(3^(k+1-1))-1/2
therefore P(k+1) is also true

by mathematical induction, P(n) is true for all natural numbers n


多口講句... MI黎講呢題算淺 =__="
如果真係唔識要俾多d心機讀書先得

When n = 1, a[1] = (5/2) * (3^0) - 1/2 = 2.
The statement is true for n = 1.

Suppose the statement is true for n = k, for some k > 1, k is a natural number
a[k] = (5/2) * (3^(k - 1)) - 1/2
When n = k + 1,
a[k + 1]
= 3 a[k] + 1
= 3 * [5/2 * (3^(k - 1)) - 1/2] + 1
= 5/2 * (3^k) - 3/2 + 1
= 5/2 * (3^k) - 1/2
The statement is true for n = k + 1.

Therefore, by mathematical induction, an=(5/2)(3^(n-1))-1/2 for all natural numbers n.