### XAT 2010 Question 81

Instructions

Questions are followed by two statements labelled as I and II. Decide if these statements are sufficient to conclusively answer the question. Choose the appropriate answer from the options given below:
A. Statement I alone is sufficient to answer the question.
B. Statement II alone is sufficient to answer the question.
C. Statement I and Statement II together are sufficient, but neither of the two alone is sufficient to answer the question.
D. Either Statement I or Statement II alone is sufficient to answer the question.
E. Both Statement I and Statement II are insufficient to answer the question

Question 81

# A sequence of positive integer is defined as $$A_{n+1}=A_{n}^{2}+1$$ for each n ≥ 0. What is the value of Greatest Common Divisor of $$A_{900}$$ and $$A_{1000}$$ ?I. $$A_{0} = 1$$II. $$A_{1} = 2$$

Solution

Expression : $$A_{n+1}=A_{n}^{2}+1$$ ---------Eqn(I)

Statement I : $$A_{0} = 1$$

Putting n = 0 in Eqn (I), => $$A_1 = A_0^2 + 1 = 1 + 1 = 2$$

Similarly, $$A_2 = A_1^2 + 1 = 4 + 1 = 5$$ and so on

We can find the values of $$A_{900}$$ and $$A_{1000}$$ and also their greatest common divisor.

Thus, statement I alone is sufficient.

Statement II : We have $$A_{1} = 2$$

In the above manner, we can determine $$A_{900}$$ and $$A_{1000}$$ and also their greatest common divisor.

Thus, statement II alone is sufficient.

$$\therefore$$ Either statement alone is sufficient.