1. a) Suppose that fis defined recursively by: f(0) =5 and f(n+1) =2 fn+5. Find A(1), A2), A3) and f(4)? b) For which positive integer n is it true that 2" >n³?

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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1. а)
Suppose that f'is defined recursively by:
f(0) = 5 and f(n+1) = 2 fn+5. Find f(1), A2), A3) and f(4)?
b)
For which positive integer n is it true that 2" >n³?
c)
Prove your answer in (b) above using mathematical induction.
d)
Give a recursive definition of the sequence {an}, n= 1, 2, 3...
if
an = 2" + 1
е)
Use your definition in (d) above to find a10, and a15
f)
Let A = {1, 2, {{1,2}}}. Find the power set P(A)
2. а)
Define the following with an example:
i)
Contradiction
ii)
Proposition
b)
Show that (p → q)^ (p→r)is a Contingency.
с)
Use quantifiers to express the statement “if somebody is
female and is a parent, then this person is someone's mother
d)
Let p and q be the propositions:
p: I will do every exercise in this book
q: I will get an 'A’ in this course
Write the following propositions using p and q and logical
connectives.
i)
For me to get an 'A’ in this course it is necessary and
sufficient that I do every exercise in this book.
ii)
Construct the truth table for your proposit
in d(i) above.
e)
i)
Write down the output Boolean expression using p, q and r
for the circuit gates below:
ii)
if p = 1, q = 0 and r= 1 find the output signal for the
circuit below:
r
Transcribed Image Text:1. а) Suppose that f'is defined recursively by: f(0) = 5 and f(n+1) = 2 fn+5. Find f(1), A2), A3) and f(4)? b) For which positive integer n is it true that 2" >n³? c) Prove your answer in (b) above using mathematical induction. d) Give a recursive definition of the sequence {an}, n= 1, 2, 3... if an = 2" + 1 е) Use your definition in (d) above to find a10, and a15 f) Let A = {1, 2, {{1,2}}}. Find the power set P(A) 2. а) Define the following with an example: i) Contradiction ii) Proposition b) Show that (p → q)^ (p→r)is a Contingency. с) Use quantifiers to express the statement “if somebody is female and is a parent, then this person is someone's mother d) Let p and q be the propositions: p: I will do every exercise in this book q: I will get an 'A’ in this course Write the following propositions using p and q and logical connectives. i) For me to get an 'A’ in this course it is necessary and sufficient that I do every exercise in this book. ii) Construct the truth table for your proposit in d(i) above. e) i) Write down the output Boolean expression using p, q and r for the circuit gates below: ii) if p = 1, q = 0 and r= 1 find the output signal for the circuit below: r
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