







Fibonacci and induction 
20100712 

From James: I'm trying to prove by induction that F(n) <= 2^(n1)
where f(1)=f(2)=1 and f(k)=f(k1)+f(k2) for k >=3 is the Fibonacci sequence Answered by Stephen La Rocque and Tyler Wood. 





A proof by induction 
20100325 

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^(n1) log x 
20100310 

From shambodeb: This is a successive differentiation problem by Leibnitz theorem
If y = x^{n1} log x ; Proof nth derivative y^{(n)} = (n1)!/x Answered by Harley Weston. 





1^3 + 2^3 + 3^3 +4^3 ... n^3 = ? 
20100129 

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 
20100112 

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 
20091002 

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 
20090602 

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 
20080905 

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 
20080711 

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 
20080623 

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 
20080317 

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 
20080317 

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 
20080314 

From Marcelo: Prove by the principle of the math induction that:
1.3.5.7....(2n1) = (2n)!/(2^n)n! Answered by Harley Weston. 





1/(1x2)+1/(2x3)+1/(3x4)...+1/(n(n+1)) 
20080220 

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 
20071215 

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 
20071127 

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 
20071031 

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 = Fn1 + Fn2
and Fn1 and Fn2 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 
20071030 

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 = 2an1
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) 
20070829 

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 
20070302 

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 
20061214 

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 
20061207 

From Carol: S(n) = 2^n > 10n+7 and n>=10 Answered by Stephen La Rocque. 





The Fibonacci sequence 
20061121 

From Ross: Let f0 = 0; f1 = 1,... be the Fibonacci sequence where for all n greater than or equal to 2 fn = fn1 + fn2. 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^(n1). Answered by Penny Nom. 





Composition of functions 
20061119 

From RJ: Let f0(x) = 2/2x 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 
20061116 

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 
20061106 

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)+2n1))/12 Answered by Penny Nom. 





Induction 
20061031 

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 
20061002 

From Zamira: i'm studying induction but i don't get how to proof that 1+2+2^2+2^3+...+2^(n1) = (2^n)  1. Answered by Penny Nom. 





Proof by induction 
20060424 

From Meshaal: Find an expression for:
13+5  7 + 9  11 + ... + (1)^(n1) * (2n1)
and prove that it is correct.
Answered by Stephen La Rocque. 





Proving a summation formula by induction 
20060419 

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 
20060409 

From Sharon: prove by induction: For every n>1, show that
2 + 7 + 12 + ...+ (5n3) = n(5n1)/2 Answered by Penny Nom. 





Proof by induction 
20060210 

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? 
20050810 

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 
20041120 

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) = 2n1 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 
20040330 

From Jose: How can you prove by mathematical induction that:
n! > n2. Answered by Penny Nom. 





Proof by induction 
20040302 

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 
20021014 

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 
20020926 

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 
20020831 

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 
20020220 

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 
20011016 

From John: Can you help me with any of these?  For any natural number n > 1, prove that
(4^{n}) / (n + 1) < [(2n)!] / [(n!)^{2}].
 For any natural number n > 1, prove that
1/sqrt(1) + 1/sqrt(2) + 1/sqrt(3) + ... + 1/sqrt(n) > sqrt(n).
 For any natural number n and any x > 0, prove that
x^{n} + x^{n  2} + x^{n  4} + ... + x^{n} >= n + 1. Answered by Penny Nom. 





Proof by induction 
20010930 

From Kyle: I'm trying to learn induction and I need to see how this done please help with this problem... 2^{0} + 2^{1} + 2^{2} +... + 2^{n} = 2^{n+1} 1 is true whenever n is a positive integer. Answered by Penny Nom. 





Harmonic numbers 
20010523 

From Leslie: The harmonic numbers H_{k}, k = 1,2,3.....are defined by H_{k} = 1 + 1/2 + 1/3....1/k I am trying to prove by mathematical induction: H_{2n} >= 1 + n/2 , whenever n is a nonnegative integer. H_{8} = H_{23} >= 1 + 3/2 Can you help? Answered by Harley Weston. 





A sequence of even terms 
20010429 

From A student: A sequence c is defined recursively as follows: c0 = 2 c1 = 4 c2 = 6 ck= 5ck3 for all integers Prove that cn is even for all integers. Answered by Leeanne Boehm and Penny Nom. 





Induction 
20000907 

From Joe Peterson: How do I prove by the principal of mathematical induction? 1.n+2.(n1)+3.(n2)+.....+(n2).3+(n1).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 
20000601 

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 
20000316 

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 
20000307 

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 
19991006 

From Polly Mackenzie: What is the difference between logic and math logic? Answered by Walter Whiteley. 





Mathematical Induction and the Derivative 
19970318 

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. 

