Recurrence

Question 1
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 .
A
198
B
148
C
226
D
312
GATE CSE 2016 SET-1   Discrete Mathematics
Question 2
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}?
A
a_{n}=a_{n-1}+2a_{n-2}
B
a_{n}=a_{n-1}+a_{n-2}
C
a_{n}=2a_{n-1}+a_{n-2}
D
a_{n}=2a_{n-1}+2a_{n-2}
GATE CSE 2016 SET-1   Discrete Mathematics
Question 3
Let a_{n} represent the number of bit strings of length n containing two consecutive 1s. What is the recurrence relation for a_{n}?
A
a_{n-2}+a_{n-1}+2^{n-2}
B
a_{n-2}+2a_{n-1}+2^{n-2}
C
2a_{n-2}+a_{n-1}+2^{n-2}
D
2a_{n-2}+2a_{n-1}+2^{n-2}
GATE CSE 2015 SET-1   Discrete Mathematics
Question 4
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. 0.5
A
i only
B
i and ii only
C
i, ii and iii only
D
i, ii, iii and iv
GATE IT 2007   Discrete Mathematics
Question 5
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?
A
\frac{1+\sqrt{1+8c}}{2c}
B
\frac{1-\sqrt{1+8c}}{2c}
C
2
D
\frac{2}{2c-1}
GATE IT 2007   Discrete Mathematics
Question 6
Let H_1, H_2, H_3, ... be harmonic numbers. Then, for n \in Z^+, \sum_{j=1}^{n} H_j can be expressed as
A
nH_{n+1} - (n + 1)
B
(n + 1)H_n - n
C
nH_n - n
D
(n + 1) H_{n+1} - (n + 1)
GATE IT 2004   Discrete Mathematics
There are 6 questions to complete.

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