Given the recurrence relation f ( n + 1 ) = 2 1 { f ( n ) + f ( n ) 9 } , where n ∈ N and f ( n ) 0 for all n ∈ N . If the limit n → ∞ lim f ( n ) exists, then determine its value.

Question

Given the recurrence relation f ( n + 1 ) = 2 1 ​ { f ( n ) + f ( n ) 9 ​ } , where n ∈ N and f ( n ) > 0 for all n ∈ N . If the limit n → ∞ lim ​ f ( n ) exists, then determine its value.

TestHub · Accepted Answer

Let n → ∞ Lt ​ f n = l ⇒ x → ∞ Lt ​ f ( n + 1 ) = l ​ Lt n → ∞ ​ f ( n + 1 ) = 2 1 ​ n → ∞ L t ​ [ f ( n ) + f ( n ) 9 ​ ] ​ ⇒ l = 2 1 ​ [ l + l 9 ​ ] ​ ​ 2 l = l l 2 + 9 ​ ⇒ 2 l 2 = l 2 + 9 ⇒ l 2 = 9 ∵ f ( n ) > 0∀ n ∈ N ∴ Lt x → ∞ ​ f ( n ) = 3 ​

Mathematics - Limits Question with Solution | TestHub

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