sequences: write set w/ array of 0s, 1s; find formula; etc

[geekinpink]

1. Let U = { m,n,o,p,q,r,s,t,u,v } A = { o,u } B = { m,o,s,t }, C = { n,p,r}, and
D = { n,o,r,t,u,v }
Represent each of the following sets by an array of zeros and ones

(a) A ? C={
(b) B ? D
(c) C ?( A ? D )
(d) ( A ? D ) ? C

2. Write a formula for the nth term of the sequence 2, 5, 8, 11, 14,….

3. Define a sequence as follows : a0 = 2, a1 = 3, an = 2an-1 – 3an-2. Give the first six terms of this sequence.

4. Let I = { 0,1}. Describe the regular subsets of I* corresponding to the following regular expressions.

(a) (01) * 10* 1
(b) 110 * 10 *
(c) ( 0 * 10 * ( 10 * 10 * )*

5. Let I = { a, b, c }. In each part below is listed a string in I* and a regular
expression over I. For each, state whether the string belongs to the regular set corresponding to the expression.

(a) abc abc*

6. Write an explicit formula for the sequence 2, 5, 8, 11, 14, 17, ……

7. Write a recursive formula for the sequence 2, 5, 7, 12, 19, 31,….

help me resolve all this question....

 

[soroban]

Re: sequence help

geekinpink!

\(\displaystyle \text{2. Write a formula for the }n^{th}\text{ term of the sequence: }\; 2, 5, 8, 11, 14, \hdots\)

\(\displaystyle a_n \;=\;3n-1\)

\(\displaystyle \text{3. Define a sequence as follows: }\:a_0 = 2,\;\;a_1 = 3,\;\;a_n \:=\:2a_{n-1} - 3a_{n-2}\)
\(\displaystyle \text{Give the first six terms of this sequence.}\)

\(\displaystyle \begin{array}{ccccc}a_0 &=& 2 \\ a_1 &=& 3 \\ a_2 &=& 2(3) - 3(2) &=& 0 \\ a_3 &=& 2(0) - 3(3) &=& \text{-}9 \\ a_4 &=& 2(\text{-}9) - 3(0) &=& \text{-}18 \\ a_5 &=& 2(\text{-}18) - 3(\text{-}9) &=& \text{-}9 \end{array}\)

\(\displaystyle \text{7. Write a recursive formula for the sequence: }\; 2, 5, 7, 12, 19, 31,\hdots\)

Each term is the sum of the preceding two terms . . .

. . . \(\displaystyle a_0 \,=\,2,\;\;a_1\,=\,5,\;\;a_n \:=\:a_{n-1} + a_{n-2}\)