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Let $a_{n+1}= 2 a_{n}+1 ({n}=0, 1, 2,\dots)$ and $a_{0}=0$. Then $a_{10}$ nearest to.

  1. $1023$
  2. $2047$
  3. $4095$
  4. $8195$
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Given that, $a_{n+1} = 2a_{n}+1; n=0,1,2,3, \dots , a_{0} = 0$

put the various values of $n$ in equation $(1)$ and observe the pattern.

  • $n=0: a_{1} = 2a_{0}+1 = 2(0)+1=1 = 2^{1}-1$
  • $n=1: a_{2} = 2a_{1}+1 = 2(1)+1=3 = 2^{2}-1$
  • $n=2: a_{3} = 2a_{2}+1 = 2(3)+1=7 = 2^{3}-1$
  • $n=3: a_{4} = 2a_{3}+1 = 2(7)+1=15 = 2^{4}-1$
  • $\vdots$
  • $n=k : a_{k+1} = 2a_{k}+1 = 2^{k+1}-1$
  • $n=9 : a_{10} = 2a_{9}+1 = 2^{10}-1 = 1024-1 = 1023$
  • Correct Answer $: \text{A}$
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