NTA Abhyas JEE Main2020MathematicsSequences and SeriesPractice
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
Options
- Aα β
- Bα + β α - β
- Cβ α
- Dα + β α - β 4
Correct answer
D. α + β α - β 4
Step-by-step solution
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