Induction Questions: Proving Statements and Solving Problems

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In summary, the conversation includes a student seeking help with two induction questions. The first question involves finding a pattern in numbers and the second question involves showing that a function is positive and its numerator is less than a given number. The student also shares their attempts at solving the problems and asks for guidance.
  • #1
moocav
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2 Induction Questions

Homework Statement


I have quite a few questions and so i just made it an image. Also attached.
http://img411.imageshack.us/img411/1002/inductionforlife.jpg" [Broken]
Only need help with questions 2 and 5 now

Oh and so far my lecturer has taught well-ordering, strong induction and simple induction. But I could only follow simple induction... the other two I'm quite clueless about >< Though tell me which method is best for each question.

Homework Equations


Refer to image

3. The Attempt at a Solution s
Question 2
I have no clue to how to start it..
All i have done is
24 = 7 + 7 + 5 + 5
25 = 5 + 5 + 5 + 5 + 5
26 = 7 + 7 + 7 + 5
27 = 7 + 5 + 5 + 5 + 5
28 = 7 + 7 + 7 + 7
29 = 7 + 7 + 5 + 5 + 5
no idea what to do now

[STRIKE]Question 3
I'm not sure if my method is correct but I've proved that when x = 0 and n = 1, x = 1 n = 2 are true. But I get stuck whilst proving n = k + 1

Let Sn be (1 + x)n >= 1 + nx
For n = 1 and x = 0, S1 =
LHS = (1+0)1 = 1
RHS = 1 + (1)(0) = 1
Therfore LHS >= RHS Hence n = 1 is true.

Assume n = k is true
Sk --> (1 + x)k >= 1 + kx

For n = k + 1, Sk+1 =

I know that I need to get to
(1 + x)k+1 >= 1 + (k+1)x

(1 + x)k >= 1 + kx
(1 + x)k(1 + x)1 >= (1 + kx)(1 + x)1 (multiplied both sides by (x + 1)
(1 + x)k+1 >= 1 + x + kx + kx2

I can see that on the RHS there is 1 + x + kx I'm not sure what to do with it... hints/help?

Question 4 (Just needs checking)

Let Sn be ƒ1 + ƒ2 + ... + ƒn = ƒn+2 -1
For n = 1, S1
LHS = ƒ1 = 1
RHS = ƒ1+2 - 1 = 2 - 1 = 1
LHS = RHS
Therefore n = 1 is true

Assume true for n = k
Sk --> ƒ1 + ƒ2 + ... + ƒk = ƒk+2 -1

For n = k + 1, Sk+1 =
RHS = ƒk+3 - 1
LHS = ƒ1 + ƒ2 + ... + ƒk + ƒk+1
= ƒk+2 -1 + ƒk+1
= ƒk+2 + ƒk+1 - 1
= ƒk+3 - 1 (should I write any reason here? if yes..what should i write?)
= RHS
Hence n = k + 1 is true
By mathematical induction Sn is true for all positive integers n.
[/STRIKE]
Question 5
Show that n/t - 1/(q+1) is positive and numerator is less than n
where t = nq + r with 0 < r < n

(get common denominator then expand and simplify)
n/t - 1/(q+1)
= n(q + 1)/[t(q+1)] - t/[t(q+1)]
= [n(q+1) - t] / [t(q+1)]
= [nq - t + n] / [t(q+1)]

t = nq + r
nq - t = -r

hence n/t - 1/(q+1)
= [n-r] / [t(q+1)]

from 0 < r < n
n > r therefore n - r > 0 (proved that numerator is positive)
and since r > 0 then n - r < n (proved that numerator is < n)

I'm not sure where to go from here

Please someone help me however you can..
Thank you in advance!
 

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  • #2
Bump (updated)
 
  • #3
It's better to post one question at a time rather than post a whole slew of them.
 
  • #4
Mark44 said:
It's better to post one question at a time rather than post a whole slew of them.

Its only two questions now ><
and they don't have to answer them all...just whichever ones they can ><
 
Last edited:

1. What is induction?

Induction is a scientific method used to make generalizations or predictions based on specific observations or data. It involves using specific examples to draw conclusions about larger patterns or principles.

2. What is the purpose of induction?

The purpose of induction is to create a hypothesis or theory that can be tested and validated through further experiments or observations. It allows scientists to make predictions and understand the underlying principles or patterns of a phenomenon.

3. How is induction different from deduction?

Induction involves using specific examples to draw general conclusions, while deduction involves using general principles to make specific predictions. Induction is based on observations and data, while deduction is based on logic and reasoning.

4. What are some examples of induction in science?

One example of induction in science is the theory of evolution, which was developed based on observations of variations in species over time. Another example is the law of gravity, which was formulated based on observations of objects falling towards the Earth.

5. What are the limitations of induction?

One limitation of induction is that it can only provide probable, not absolute, conclusions. Additionally, the conclusions drawn through induction may be biased by the specific examples or observations used. It is important to use caution and further testing when making generalizations through induction.

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