Let f : R → R be a function which satisfies f x + y = f x + f y , ∀ x , y ∈ R . If f 1 = 2 and g n = ∑ k = 1 n - 1 f k , n ∈ N then the value of n , for which g n = 20 , is

Question

Let f : R → R be a function which satisfies f x + y = f x + f y , ∀ x , y ∈ R . If f 1 = 2 and g n = ∑ k = 1 n - 1 f k , n ∈ N then the value of n , for which g n = 20 , is, 5, 20, 4, 9

MARKS App · Accepted Answer

∵ f x + y = f x + f y ∴ f x = k x , ∀ x ∈ R Now f 1 = 2 ⇒ k = 2 ∴ f x = 2 x , ∀ x ∈ R g n = f 1 + f 2 + … … . . + f n - 1 = 2 1 + 2 + 3 + … … + n - 1 = 2 · n - 1 2 n = 20 ⇒ n 2 - n - 20 = 0 ⇒ n = 5 .

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