The value of $6+\log _{\frac{3}{2}}\left(\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \sqrt{4-\frac{1}{3 \sqrt{2}} \ldots}}}\right)$ is

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Sequences and Series

The value of

6 + lo g_{\frac{3}{2}} (\frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} \dots)

Answer: 4

Solution

Let

y = 4 - \frac{1}{3 2} 4 - \frac{1}{3 2} 4 - \dots

. Then

y = 4 - \frac{y}{3 2}

, so

y^{2} = 4 - \frac{y}{3 2}

. This gives

y^{2} + \frac{y}{3 2} - 4 = 0

. Solving:

y = \frac{- \frac{1}{3 2} + \frac{1}{18} + 16}{2} = \frac{8}{3 2}

(taking the positive root).

The expression becomes

6 + lo g_{3/2} (\frac{1}{3 2} \cdot \frac{8}{3 2}) = 6 + lo g_{3/2} (\frac{4}{9}) = 6 + lo g_{3/2} (2^{2} \cdot 3^{- 2}) = 6 - 2 = 4

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Integer answer

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