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mathematical induction

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23 items are filed under this topic.
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
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.
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.
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.
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?)

Answered by Penny Nom and Victoria West.
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
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.
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.
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.
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?

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


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.
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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