Let α and β are two positive roots of x 2 - 2 a x + a b = 0 where 0 b a , then the value of S n = 1 + 2 b a + 3 b a 2 + . . . . . + n b a n - 1 , ∀ n ∈ N cannot exceed

Question

Let α and β are two positive roots of x 2 - 2 a x + a b = 0 where 0 b a , then the value of S n = 1 + 2 b a + 3 b a 2 + . . . . . + n b a n - 1 , ∀ n ∈ N cannot exceed, α β, α + β α - β, β α, α + β α - β 4

MARKS App · Accepted Answer

S n S ∞ ⇒ S n 1 + 2 b a + 3 b a 2 + . . . . upto ∞ ⇒ S n 1 1 - b a + 1 × b a 1 - b a 2 = 1 1 - b a 2 = a a - b 2 ⇒ S n 4 a 2 4 a 2 - 4 a b 2 Now, α + β = 2 a and α β = a b ⇒ S n α + β 2 α + β 2 - 4 α β 2 = α + β 2 α - β 2 2 ⇒ S n α + β α - β 4

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