Let a1, a2, a3 ........ be in harmonic progression with a1 = 5 and a20 = 25. The least positive integer n for which an

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Mathematics

Harmonic Progression and Its Properties

Question

Let a₁, a₂, a₃ ........ be in harmonic progression with a₁ = 5 and a₂₀ = 25. The least positive integer n for which a_n < 0 is

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$\frac{1}{a_{1}}, \frac{1}{a_{2}}, \frac{1}{a_{3}} . ....... \frac{1}{a_{20}}$ Arithmetic Progression

$\Rightarrow \frac{1}{5}, \frac{1}{a_{2}}, \frac{1}{a_{3}} . ....... \frac{1}{25}$ Arithmetic Progression

$∴ \frac{1}{25} = \frac{1}{5} + 19 d \Rightarrow d = \frac{- 4}{25 .19}$ Let n^th term of Arithmetic progression $\frac{1}{a_{n}}$ is negative

$∴ \frac{1}{a_{n}} = \frac{1}{5} - \frac{4 (n - 1)}{25 \cdot 19} < 0 \Rightarrow$ (n – 1) 4 > 19 · 5

$\Rightarrow n > \frac{99}{4}$ (less than 25) $\Rightarrow$ n = 25. Ans.]

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