Question

馬卡巴卡

請問這題要怎麼解🥺🙏🙏

<An> {0₁-1
1721) : Un =
並說明之
- An= 4-an²+ (n=2)
3-An-1

Answer 1

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smallroad

已知 a1 = 1
a[n] = (4 - a[n-1] ) / (3 - a[n-1] ), n >= 2

試帶入n = 2
a[2] = (4 - a[2-1] ) / (3 - a[2-1] )
-> a[2] = (4 - a[1] ) / (3 - a[1] )
-> a[2] = (4 - 1 ) / (3 - 1 )
-> a[2] = 3 / 2

同理再帶入n = 3
a[3] = (4 - a[3-1] ) / (3 - a[3-1] )
-> a[3] = (4 - a[2] ) / (3 - a[2] )
-> a[3] = (4 - 3/2 ) / (3 - 3/2 )
-> a[3] = 5 / 3

帶入n = 4
a[4] = (4 - a[4-1] ) / (3 - a[4-1] )
-> a[4] = (4 - a[3] ) / (3 - a[3] )
-> a[4] = (4 - 5/3 ) / (3 - 5/3 )
-> a[4] = 7 / 4

可發現 a[1]、a[2]、a[3]、a[4]依序是 1/1、3/2、5/3、7/4
依照此規律可推論 an = (2n - 1) / (n)

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