Recurrence


Question 1
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?
A
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{2} \right )^n
B
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n - \left ( \frac{1-\sqrt{5}}{3} \right )^n
C
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n+\left ( \frac{1-\sqrt{5}}{3} \right )^n
D
L_n=\left ( \frac{1+\sqrt{5}}{2} \right )^n- \left ( \frac{1-\sqrt{5}}{2} \right )^n
GATE CSE 2023   Discrete Mathematics
Question 2
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?
MSQ
A
f(2^n-1)=2^n-1
B
f(2^n)=1
C
f(5 \dot 2^n)=2^{n+1}+1
D
f(2^n+1)=2^n+1
GATE CSE 2022   Discrete Mathematics


Question 3
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 4
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 5
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


There are 5 questions to complete.

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