f ( x ) = 2 x 1 + 9 x 2 f(x)=\frac{2 x}{\sqrt{1+9 x^2}} f ( x ) = 1 + 9 x 2 2 x
( f ∘ f ) ( x ) = 2 f ( x ) 1 + 9 ( f ( x ) ) 2 = 4 x 1 + 9 x 2 1 + 9 × 4 x 2 1 + 9 x 2 = 4 x 1 + 45 x 2 (f \circ f)(x)=\frac{2 f(x)}{\sqrt{1+9(f(x))^2}}=\frac{\frac{4 x}{\sqrt{1+9 x^2}}}{\sqrt{1+9 \times \frac{4 x^2}{1+9 x^2}}}=\frac{4 x}{\sqrt{1+45 x^2}} ( f ∘ f ) ( x ) = 1 + 9 ( f ( x ) ) 2 2 f ( x ) = 1 + 9 × 1 + 9 x 2 4 x 2 1 + 9 x 2 4 x = 1 + 45 x 2 4 x
( f ∘ f ∘ f ) ( x ) = 4 × 2 x 1 + 9 x 2 1 + 45 × 4 x 2 1 + 9 x 2 = 8 x 1 + 21 × 9 x 2 (f \circ f \circ f)(x)=\frac{4 \times \frac{2 x}{\sqrt{1+9 x^2}}}{\sqrt{1+45 \times \frac{4 x^2}{1+9 x^2}}}=\frac{8 x}{\sqrt{1+21 \times 9 x^2}} ( f ∘ f ∘ f ) ( x ) = 1 + 45 × 1 + 9 x 2 4 x 2 4 × 1 + 9 x 2 2 x = 1 + 21 × 9 x 2 8 x
( f ∘ f ∘ f ∘ f ) ( x ) = 16 x 1 + 85 × 9 x 2 (f \circ f \circ f \circ f)(x)=\frac{16 x}{\sqrt{1+85 \times 9 x^2}} ( f ∘ f ∘ f ∘ f ) ( x ) = 1 + 85 × 9 x 2 16 x
⇒ α \Rightarrow \alpha ⇒ α is 10 th 10^{\text {th }} 1 0 th term of 1 , 5 , 21 , 85 , … α 1,5,21,85, \ldots \alpha 1 , 5 , 21 , 85 , … α is 10 th 10^{\text {th }} 1 0 th term of
a m p ; ( 2 1 ) 2 − 1 3 , ( 2 2 ) 2 − 1 3 , ( 2 3 ) 2 − 1 3 , ( 2 4 ) 2 − 1 3 , … ⇒ a m p ; α = ( 2 10 ) 2 − 1 3 ⇒ a m p ; 3 α + 1 = 2 10 = 1024 \begin{aligned}& \frac{\left(2^1\right)^2-1}{3}, \frac{\left(2^2\right)^2-1}{3}, \frac{\left(2^3\right)^2-1}{3}, \frac{\left(2^4\right)^2-1}{3}, \ldots \\\Rightarrow \quad & \alpha=\frac{\left(2^{10}\right)^2-1}{3} \\\Rightarrow \quad & \sqrt{3 \alpha+1}=2^{10}=1024\end{aligned} ⇒ ⇒ am p ; 3 ( 2 1 ) 2 − 1 , 3 ( 2 2 ) 2 − 1 , 3 ( 2 3 ) 2 − 1 , 3 ( 2 4 ) 2 − 1 , … am p ; α = 3 ( 2 10 ) 2 − 1 am p ; 3 α + 1 = 2 10 = 1024