Recurrence
Q1.
Consider the sequence \langle x_n \rangle , \: n \geq 0 defined by the recurrence relation x_{n+1} = c . x^2_n -2, where c > 0. Suppose there exists a non-empty, open interval (a, b) such that for all x_0 satisfying a \lt x_0 \lt b, the sequence converges to a limit. The sequence converges to the value?Q2.
Consider the sequence \left \langle x_n \right \rangle,\; n \geq 0 defined by the recurrence relation x_{n + 1} = c \cdot (x_n)^2 - 2, where c > 0. For which of the following values of c, does there exist a non-empty open interval (a, b) such that the sequence x_n converges for all x_0 satisfying a < x_0 < b? i. 0.25 ii. 0.35 iii. 0.45 iv. 0.5Q3.
Let a_{n} be the number of n-bit strings that do NOT contain two consecutive 1s. Which one of the following is the recurrence relation for a_{n}?Q4.
The Lucas sequence L_n is defined by the recurrence relation: L_n=L_{n-1}+L_{n-2}, \; for \; n\geq 3, with L_1=1 \; and \; L_2=3 Which one of the options given is TRUE?Q5.
Consider the recurrence relation a_{1}=8, a_{n}=6n^{2}+2n+a_{n-1}. Let a_{99}= K \times 10^{4}. The value of K is .Q6.
Consider the following recurrence: \begin{aligned} f(1)&=1; \\ f(2n)&=2f(n)-1, & \text{for }n \geq 1; \\ f(2n+1)&=2f(n)+1, & \text{for }n \geq 1. \end{aligned}Then, which of the following statements is/are TRUE?MSQQ7.
Let H_1, H_2, H_3, ... be harmonic numbers. Then, for n \in Z^+, \sum_{j=1}^{n} H_j can be expressed asQ8.
Let a_{n} represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for a_{n}?