Question

The sequence $\left( {{x_n},n \geqslant 1} \right)$ is defined by ${x_1} = 0$ and the ${x_{n + 1}} = 5{x_n} + \sqrt {24{x_n}^2 + 1} $ for all \[n \geqslant 1\]. Then all ${x_n}$ are
A) Negative integers
B) Positive integers
C) Rational numbers
D) None of these

Answer 1

Hint:
Find the next terms of the sequence by substituting different values of n. Observe the pattern of sequence. Since, the sequence is increasing and positive. All the terms of $x_n$ will be positive.

Complete step by step solution:
We have ${x_1} = 0$. Find the next term of the sequence by substituting $n = 1$ in the general term ${x_{n + 1}} = 5{x_n} + \sqrt {24{x_n}^2 + 1} $
$
  {x_{1 + 1}} = 5{x_1} + \sqrt {24{x_1}^2 + 1} \\
  {x_2} = 5\left( 0 \right) + \sqrt {24\left( 0 \right) + 1} \\
  {x_2} = 1 \\
$
Similarly, find next term by substituting $n = 2$ in the general term ${x_{n + 1}} = 5{x_n} + \sqrt {24{x_n}^2 + 1} $
$
  {x_{2 + 1}} = 5{x_2} + \sqrt {24{x_2}^2 + 1} \\
  {x_3} = 5\left( 1 \right) + \sqrt {24\left( 1 \right) + 1} \\
  {x_3} = 5 + \sqrt {25} \\
  {x_3} = 5 + 5 \\
  {x_3} = 10 \\
 $
Similarly, find next term by substituting $n = 3$ in the general term ${x_{n + 1}} = 5{x_n} + \sqrt {24{x_n}^2 + 1} $
$
  {x_{3 + 1}} = 5{x_3} + \sqrt {24{x_3}^2 + 1} \\
  {x_4} = 5\left( {10} \right) + \sqrt {24\left( {10} \right) + 1} \\
  {x_4} = 50 + \sqrt {241} \\
$
We observe that the sequence is an increasing sequence and terms are positive.

Hence, option B is the correct option.

Note:
When the value of terms increases as the value of $n$ increases, then the sequence ${x_n}$ is said to be an increasing sequence. Similarly, When the value of terms decreases as the value of $n$ increases, then the sequence ${x_n}$ is said to be a decreasing sequence.