







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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 





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. 

