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induction

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Fibonacci and induction 2010-07-12
From James:
I'm trying to prove by induction that F(n) <= 2^(n-1) where f(1)=f(2)=1 and f(k)=f(k-1)+f(k-2) for k >=3 is the Fibonacci sequence
Answered by Stephen La Rocque and Tyler Wood.
A proof by induction 2010-03-25
From SAMUEL:
use mathematical induction to proof that each statement is true for every positve integer n 1/1.2+1/2.3+1/3.4+......1/n(n+1)=n/n+1
Answered by Robert Dawson.
The nth derivative of x^(n-1) log x 2010-03-10
From shambodeb:
This is a successive differentiation problem by Leibnitz theorem

If y = xn-1 log x ; Proof nth derivative y(n) = (n-1)!/x

Answered by Harley Weston.
1^3 + 2^3 + 3^3 +4^3 ... n^3 = ? 2010-01-29
From ireimaima:
Hi..
Can u please help me with this question.. I find that when i test eg: n=2 for n (n+1) /4, it seems that it does not giving me the right answer of 1^3 + 2^3 = 9 but 3/2... i'm confuse..can u please help me..thanks so much

Prove that: 1^3 + 2^3 + 3^3 +4^3..n^3 = n (n+1) /4

Answered by Penny Nom.
A proof by induction 2010-01-12
From Bhavya:
Prove by induction that if Xi >= 0 for all i, then

(Summation Xi from 1 to n)^2 >= Summation Xi^2 from 1 to n

Answered by Penny Nom.
Prove by induction 2009-10-02
From Anonymous:
How can you prove the following by induction:

Any fraction (A / B), where 0 < (A / B) < 1, can be expressed as a finite sum
(1 / c(1)) + (1 / c(2)) + (1 / c(3)) + ... + (1 / c(k)),
where c(1), c(2), ..., c(k) are natural numbers greater than 0.

[ex. (20 / 99) = (1 / 9) + (1 / 11)]

Answered by Claude Tardif.
Selecting 3 people from 4 2009-06-02
From muhammadibeaheem:
Use mathematical induction to prove that for all integers n≥1, is divisible by 3.

Question 2; A club consists of four members.How many sample points are in the sample space when three officers; president, secretary and treasurer, are to be chosen?

Answered by Penny Nom.
Mathematical induction 2008-09-05
From James:
I need to prove a problem by induction regarding the Triangle Inequality. The problem is

abs(a1 + a2 +...+an) <= abs(a1) + abs(a2) +...+ abs(an).

Answered by Victoria West.
Mathematical induction 2008-07-11
From lyn:
can you give me a basic example of a mathematical induction
Answered by Harley Weston.
The sum of the digits of a number 2008-06-23
From Ben:
Question: Using mathematical induction, prove that if the sum of the digits of a number is divisible by three, then the number itself is also divisible by 3.
Answered by Penny Nom.
n^3/3 + n^5/5 + 7n/15 is an integer 2008-03-17
From John:
Prove: For all n in Natural Numbers ( n > 1 ),
n^3/3 + n^5/5 + 7n/15 is an integer

Answered by Stephen La Rocque.
2^n > n^2 for n> 4 where n is a natural number 2008-03-17
From John:
I've been asked to prove this:
2^n > n^2 for n> 4 and n is a natural number

Answered by Penny Nom.
Induction 2008-03-14
From Marcelo:
Prove by the principle of the math induction that:

1.3.5.7....(2n-1) = (2n)!/(2^n)n!

Answered by Harley Weston.
1/(1x2)+1/(2x3)+1/(3x4)...+1/(n(n+1)) 2008-02-20
From hossun:
Find a formula for 1/(1x2)+1/(2x3)+1/(3x4)...+1/(n(n+1)) by examining the values of this expression for small values of n. Use mathematical induction to prove your result.
Answered by Stephen La Rocque.
The Principle of Mathematical Induction 2007-12-15
From iris:
we have some confusion in our problem. Please help us. We would like to know "the principle of mathematical induction"
(i) for n=1, p(1) is true.
(ii) assume that for n=k>=1, p(k) is true we have to prove p(k+1) is true. Here (Is n=k>=1 true? or Is n=k.1 true?)
Thanks.

Answered by Penny Nom and Victoria West.
Mathematical induction 2007-11-27
From Angels:
Please help! Prove the formula for every positive integer
1^3+2^3+3^3+4^3+...+n^3=n^2((n+1)^2/4)

Answered by Harley Weston.
A faulty induction argument 2007-10-31
From snehal:
Find the problem in the following argument. Try to give another example that illustrates the same problem.
Claim: All Fibonacci numbers are even.

Proof: We will use strong induction. Let P(n) be the proposition that Fn is even.
Base case: F0 = 0 is even, so P(0) is true.

Inductive step: Assume P(0); : : : ; P(n - 1) to prove P(n): Now Fn = Fn-1 + Fn-2 and Fn-1 and Fn-2 are both even by assumptions P(n - 1) and P(n - 2); so Fn is also even. By induction, all Fibonacci numbers are even.

Answered by Stephen La Rocque and Claude Tardif.
Subsets of a set 2007-10-30
From Snehal:
1. Let an denote the number of subsets of f{1,2, 3.... n}including the empty set and the set itself.)
a) Show an = 2an-1
b) Guess a formula for the value of an and use induction to prove you are right

Answered by Stephen La Rocque.
Induction problem (divisible by 11) 2007-08-29
From James:
Show that 27 * (23 ^ n) + 17 * (10 )^ (2n) is divisible by 11 for all positive integers n.
Answered by Stephen La Rocque and Penny Nom.
Mathematical induction 2007-03-02
From Suud:
Hello sir/ madam I am really confused about this topic, and i am unable to understand it well. So please help me! I need to send me, clear, detailed and main notes about the principle of mathematical Induction, proofs, and applications. And I would be pleased if you sent me, some solved problems for more clarification and understanding. I would like to appreciate your help! Thank You!
Answered by Haley Ess.
cos(n)pi = (-1)^n 2006-12-14
From Idrees:
How can I prove the following: cos(n)pi = (-1)^n
Answered by Steve La Rocque.
The proof of inequality by mathematical induction 2006-12-07
From Carol:
S(n) = 2^n > 10n+7 and n>=10
Answered by Stephen La Rocque.
The Fibonacci sequence 2006-11-21
From Ross:
Let f0 = 0; f1 = 1,... be the Fibonacci sequence where for all n greater than or equal to 2 fn = fn-1 + fn-2. Let Q = (1+square root of 5)/2. Show that for all positive n greater than or equal to 0, fn less than or equal to Q^(n-1).
Answered by Penny Nom.
Composition of functions 2006-11-19
From RJ:
Let f0(x) = 2/2-x and fn+1 = f0 o fn for n greater than or equal to 0. Find a formula for fn and prove it by mathematical induction. Recall that o represents function composition. i.e., (f o g)(x) = f(g(x)).
Answered by Stephen La Rocque.
Induction 2006-11-16
From John:
Find a formula for
1/(1x3)+1/(2x4)+1/(3x5)...+1/(n(n+2))
by examining the values of this expression for small values of n. Use mathematical induction to prove your result.

Answered by Penny Nom.
A proof by induction 2006-11-06
From Zamira:
i have a problem with this mathematical induction: (1^5)+(2^5)+(3^5)+...+(n^5) = ((n^2)*((n+1)^2)*((2n^2)+2n-1))/12
Answered by Penny Nom.
Induction 2006-10-31
From Ross:
Suppose that A and B are square matrices with the property AB= BA. Show that AB^n = B^n A for every positive integer n.
Answered by Stephen La Rocque and Penny Nom.
A proof by induction 2006-10-02
From Zamira:
i'm studying induction but i don't get how to proof that 1+2+2^2+2^3+...+2^(n-1) = (2^n) - 1.
Answered by Penny Nom.
Proof by induction 2006-04-24
From Meshaal:
Find an expression for: 1-3+5 - 7 + 9 - 11 + ... + (-1)^(n-1) * (2n-1) and prove that it is correct.
Answered by Stephen La Rocque.
Proving a summation formula by induction 2006-04-19
From Sharon:
Prove by induction that the sum of all values 2^i from i=1 to n equals 2^(n+1) - 2 for n > 1.
Answered by Stephen La Rocque.
A proof by induction 2006-04-09
From Sharon:
prove by induction: For every n>1, show that
2 + 7 + 12 + ...+ (5n-3) = n(5n-1)/2

Answered by Penny Nom.
Proof by induction 2006-02-10
From Victoria:

how do i prove by induction on n that
n
Σ 1/i(i+1) = n/(n+1)
i=1

for all positive integers n


Answered by Penny Nom.
Proof by induction? 2005-08-10
From Peter:

I am a lecturer and am having a problem with the following Proof by
Induction.

If

(N x N x N x N) + (4 x N x N x N) + (3 x N x N) + (N) = -4000

Prove that N is even!


Answered by Chris Fisher and Penny Nom.
Proof by induction 2004-11-20
From Vic:
Problem: Find the first 4 terms and the nth term of the infinite sequence defined recursively as follows:

a(1) = 3 and a(k+1) = 2a(k) for k -> 1.

Note: Quantities in brackets are subscripts
-> means 'equal to or greater than'.

Using the recursive formula, the first 4 terms are; a(1) = 3, a(2) = 6, a(3) = 12, a(4) = 24

The nth term a(n) = 2n-1 x 3 (equation 1)

Equation 1 must be proven using mathematical induction. This is where I am having a problem.

Answered by Penny Nom.
n! > n^2 2004-03-30
From Jose:
How can you prove by mathematical induction that:

n! > n2.

Answered by Penny Nom.
Proof by induction 2004-03-02
From Chris:
I need some help of how to solve the problem

"use the principle of mathematical induction to prove that the following are true for all positive integers"

cos(n x pi + X) = (-1)^n cosX

any help would be appreciated

Answered by Penny Nom.
A functional equation 2002-10-14
From Rob:
Let f be a function whose domain is a set of all positive integers and whose range is a subset of the set of all positive integers with these conditions: a) f(n+1)>f(n)

b) f(f(n))=3(n)

Answered by Claude Tardif.
Proof by induction 2002-09-26
From Pooh:
Use induction to show that

1 2 + 2 2 + .....+n 2 = (n 3)/3 + (n 2)/2 + n/6

Answered by Paul Betts.
Proof by induction 2002-08-31
From Tabius:
Use mathematical induction to prove that the following formulae are true for all positive integers:

a) 1 + 3 + 5+...+(2n - 1) = n 2

b) 2 n > n.


Answered by Penny Nom.
Proof by induction 2002-02-20
From Tamaswati:
How do I prove the assertion that "the determinant of an upper triangular matrix is the product of the diagonal entries" by mathematical induction? (Before I check this assertion for a few values of n how do I rephrase the assertion slightly so that n appears explicitly in the assertion?)
Answered by Penny Nom.
Proof by induction 2001-10-16
From John:
Can you help me with any of these?
  1. For any natural number n > 1, prove that

    (4n) / (n + 1) < [(2n)!] / [(n!)2].

  2. For any natural number n > 1, prove that

    1/sqrt(1) + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(n) > sqrt(n).

  3. For any natural number n and any x > 0, prove that

    xn + xn - 2 + xn - 4 + ... + x-n >= n + 1.

Answered by Penny Nom.
Proof by induction 2001-09-30
From Kyle:
I'm trying to learn induction and I need to see how this done please help with this problem...

20 + 21 + 22 +... + 2n = 2n+1 -1 is true whenever n is a positive integer.


Answered by Penny Nom.
Harmonic numbers 2001-05-23
From Leslie:
The harmonic numbers Hk, k = 1,2,3.....are defined by Hk = 1 + 1/2 + 1/3....1/k

I am trying to prove by mathematical induction:

H2n >= 1 + n/2 , whenever n is a nonnegative integer.

H8 = H23 >= 1 + 3/2

Can you help?


Answered by Harley Weston.
A sequence of even terms 2001-04-29
From A student:
A sequence c is defined recursively as follows:

c0 = 2
c1 = 4
c2 = 6

ck= 5ck-3 for all integers

Prove that cn is even for all integers.


Answered by Leeanne Boehm and Penny Nom.
Induction 2000-09-07
From Joe Peterson:
How do I prove by the principal of mathematical induction?
1.n+2.(n-1)+3.(n-2)+.....+(n-2).3+(n-1).2+n.1=(n(n+1)(n+2))/6

Answered by Paul Betts.
1+4+9+16+...n^2 = n(n+1)(2n+1)/6 2000-06-01
From Shamus O'Toole:
How do you derive that for the series 1+4+9+16+25.. that

S(n)=(n(n+1)(2n+1))/6


Answered by Penny Nom.
Induction 2000-03-16
From William Tsang:
I am trying to prove a induction question

Sigam r=1 n (2r -1)cube = n square (2 n square - 1)


Answered by Harley Weston.
Mathematical deduction and mathematical induction 2000-03-07
From Espera Pax:
What are mathematical deduction and mathematical induction, and what is the difference between them?
Answered by Harley Weston.
Logic and mathematical logic 1999-10-06
From Polly Mackenzie:
What is the difference between logic and math logic?
Answered by Walter Whiteley.
Mathematical Induction and the Derivative 1997-03-18
From Shuling Chong:
"Obtain a formula for the nth derivative of the product of two functions, and prove the formula by induction on n."

Any educated tries are appreciated.
Answered by Penny Nom.

 
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