A Weird Infinity Fractional Sequence ~ Inspired by Sourasish K (Part 2)

Probability Level 3

1 4 + 3 4 ⋅ 4 + 3 ⋅ 5 4 ⋅ 4 ⋅ 8 + 3 ⋅ 5 ⋅ 7 4 ⋅ 4 ⋅ 8 ⋅ 12 + ⋯ = \large \dfrac{1}{4} + \dfrac{3}{4 \cdot 4} + \dfrac{3 \cdot 5}{4 \cdot 4 \cdot 8} + \dfrac{3 \cdot 5 \cdot 7 }{4 \cdot 4 \cdot 8 \cdot 12 } + \cdots = \,

  • A. 2 \quad 2
  • B. 1 2 \quad \dfrac{1}{\sqrt{2}}
  • C. 1 2 \quad \dfrac{1}{2}
  • D. 2 2 \quad \dfrac{\sqrt{2}}{2}
  • E. 4 2 \quad \dfrac{4}{2}
  • F. 2 4 \quad \dfrac{2}{4}

Inspiration

Try another problem on my set! Let's Practice
B and D are correct C and F are correct A and E are correct

Fidel Simanjuntak by Fidel Simanjuntak , 19, Indonesia

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2 solutions

Ankit Kumar Jain
May 22, 2017

( 1 − x ) − n = 1 + n x + n ( n + 1 ) 2 ! x 2 + ⋯ + n ( n + 1 ) ( n + 2 ) ⋯ ( n + r − 1 ) r ! x r + ⋯ ∞ terms (1-x)^{-n}=1+nx+\dfrac{n(n+1)}{2!}x^2+\cdots+\dfrac{n(n+1)(n+2)\cdots (n+r-1)}{r!}x^{r}+\cdots\infty\quad\text{terms}

Substitute x = 1 2 , n = 3 2 \boxed{x=\dfrac12 , n = \dfrac32}

⇒ 1 + 3 4 + 3 ⋅ 5 4 ⋅ 8 + 3 ⋅ 5 ⋅ 7 4 ⋅ 8 ⋅ 12 + ⋯ = 2 2 \Rightarrow 1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\cdots=2\sqrt{2}

Multiplying throughout by 1 4 \dfrac14 ,

⇒ 1 4 + 3 4 ⋅ 4 + 3 ⋅ 5 4 ⋅ 4 ⋅ 8 + 3 ⋅ 5 ⋅ 7 4 ⋅ 4 ⋅ 8 ⋅ 12 + ⋯ = 1 2 \Rightarrow \dfrac14+\dfrac3{4\cdot4}+\dfrac{3\cdot5}{4\cdot4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot4\cdot8\cdot12}+\cdots = \boxed{\dfrac1{\sqrt{2}}}

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I'm not thinking but i just look the option from a-f the same answer is b and d so the answer is b and d are correct(:

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