What is Mathematical induction: Definition and 213 Discussions
Mathematical induction is a mathematical proof technique. It is essentially used to prove that a statement P(n) holds for every natural number n = 0, 1, 2, 3, . . . ; that is, the overall statement is a sequence of infinitely many cases P(0), P(1), P(2), P(3), . . . . Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder:
Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step).
A proof by induction consists of two cases. The first, the base case (or basis), proves the statement for n = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that if the statement holds for any given case n = k, then it must also hold for the next case n = k + 1. These two steps establish that the statement holds for every natural number n. The base case does not necessarily begin with n = 0, but often with n = 1, and possibly with any fixed natural number n = N, establishing the truth of the statement for all natural numbers n ≥ N.
The method can be extended to prove statements about more general well-founded structures, such as trees; this generalization, known as structural induction, is used in mathematical logic and computer science. Mathematical induction in this extended sense is closely related to recursion. Mathematical induction is an inference rule used in formal proofs, and in some form is the foundation of all correctness proofs for computer programs.Although its name may suggest otherwise, mathematical induction should not be confused with inductive reasoning as used in philosophy (see Problem of induction). The mathematical method examines infinitely many cases to prove a general statement, but does so by a finite chain of deductive reasoning involving the variable n, which can take infinitely many values.
Hello,
I have a question about inductive reasoning...
Earlier this week my intro proofs class went over the logical structure of induction, and an example.
The example was a proof of \Sigmai = n*(n + 1)/2
My main issue is the assumption that "p(k)" is true. What if it's not? I asked this in...
I'm having a bit of trouble understanding why the principle of induction is included as one of Peano's axioms. It seems like it should not be independent of the others. Obviously it can be stated as:
If a predicate P is true only of natural numbers, P(0) is true, and also P(n)\rightarrow...
Hi. I am learning mathematical induction on my own and I came across this problem:
Homework Statement
Prove:
1*4 + 4*7 + 7*10 + ... + (3n - 2)(3n + 1) = n(n + 1)²2. The attempt at a solution
Quick test for n=1:
(3 -2)(3 + 1) = 1(1 + 1)²
4 = 4
Alright, so I rewrite this with, on the left...
Homework Statement
Prove that
11^n - 4^n
is a multiple of 7
Homework Equations
N/AThe Attempt at a Solution
I substituted k+1 in for n and simplified to get
11(11^k)-4(4^k)
but after this point I get stuck. Any help would be appreciated.
Homework Statement
Summation of i(i + 1) (with i going from i = 2 to i = n-1) = n(n-1)(n=1) / 3
a. Write P(2). Is P(2) true?
b. Write P(k)
c. Write P(k+1)
d. Prove by mathematical induction that the formula holds true for all integers
n \geq 2
Homework Equations...
I was working on my homework and I did two of the mathematical induction problems before this one and they were super easy but I must be forgetting something because I just can't seem to solve this one.
2+7+12+17...+(5n-3) = (n/2)(5n-1)
So I know that step 1 is to prove that it works with...
so the problem is 3+7+11+15+...+(4n-1) = n(2n+1)
so I know that step 1 is to plug in 1 for the right side and check that it equals three...
3=1(2(1)+1) and yes it equals 3
Then I know that you assume that 3+7+11+15+...+(4n-1) = n(2n+1)
The next step is where I get confused I know that...
Homework Statement
For each n\,\in\,\mathbb{N}, let p_n(x)\,\in\,\mathbb{Z}_2[x] be the polynomial
1\,+\,x\,+\,\cdots\,x^{n\,-\,1}\,+\,x^n
Use mathematical induction to prove that
p_n(x)\,\cdot\,p_n(x)\,=\,1\,+\,x^2\,+\,\cdots\,+x^{2n\,-\,2}\,+\,x^{2n}Homework Equations
Induction steps...
Homework Statement
Prove that (n + 1)n - 1 < nn for n ∈ Z+. [Hint: Induction is suggested. Write out the induction statement explicitly. Make one side of the inequality look like your induction hypothesis.]
Homework Equations
The Attempt at a Solution
^ That's what I have so far. I'm good...
Homework Statement
Prove: 1+2(2+3+4+···+n)+(n+1)=(n+1)2-1 for all n within NHomework Equations
PMI
The Attempt at a Solution
I tried applying PMI starting with the basis step, allowing n=1, assuming the equation would be true. However 1+2(1)+(1+1)=?=(1+1)2-1 yields 5=?=3 which is false. The...
Homework Statement
Use mathematical induction to prove the following statement.
n \geq 2,~ \sum_{k=1}^{n} \frac{1}{k^2}~<~1-\frac{1}{n}
Homework Equations
The Attempt at a Solution
This is the third problem I've done of this type, so I am by no means an expert. So this attempt may be...
In the context of Set theory and Relations why do we use mathematical induction. Is there any deep relation between all these concepts or mathematical induction is only a separate concepts introduced in the textbooks after Sets and Relation ; Functions ; and then Mathematical induction.
Homework Statement
The sequence a0 -> n is defined by ai = b+i*c. Prove by induction on n that the sum of the terms in the sequence is (n+1)(a0 + an)/2.Homework Equations
The Attempt at a Solution
I defined predicate P(n) as (n+1)(a0+an)/2.
My goal is P(n+1), which is (n+2)(a0+an+1)/2. I...
Homework Statement
Prove that \frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}+...+\frac{1}{\sqrt{n}}\geq\sqrt{n} for all n \in N
Homework Equations
The Attempt at a Solution
p(1): \frac{1}{\sqrt{1}} = \frac{1}{1} = 1 = \sqrt{1} \geq \sqrt{1}
Let \frac{1}{\sqrt{1}} + \frac{1}{\sqrt{2}} +...+...
Homework Statement
Consider the sequence of real numbers x1, x2, x3,... defi ned by the relations x1 = 1
and xn+1 =\sqrt{1 + 2xn}
1. Use mathematical induction to show that xn+1 > xn
for all n \geq 1.
The Attempt at a Solution
I'm a bit thrown off by this question because it...
Hey guys, this is a problem given to us by our professor in one of the worksheets. I would like an opinion to see if this proof is valid
Homework Statement
Use mathematical induction to prove that 3^{n}+7^{n}-2 is divisible by 8 for \foralln\inZ^{+}Homework Equations
Base step:
n=0...
Im practicing inductions on my own and I am getting stuck on one in particular...
S_n = \frac{3(3^n-1)}{2} for a_n = 3^n
I know that
S_1 = 3
when you plug 1 into the equation, because it is the first term in the sequence
Therefore,
S_1 = a_1 = 3
I need to prove then
S_{n+1} =...
Hi,
I'm trying to learn mathematical induction for proving inequalities, but there is just one step I cannot get past: finding another inequality that is added to the inductive hypothesis.
For example, in this problem:
Prove for all positive integers (n >= 1), prove 3^n + 2 >= 3n.
I...
Homework Statement
Let f(n + 1) = 3f(n) + 8, with f(1) = 11. Prove by induction that f(n) = 5 x 3^n - 4.Homework Equations
The Attempt at a Solution
I don't even know where to start! Any help would be appreciated. Thanks. :-)
Homework Statement
What is wrong with my solution?...
I don't quite understand where do I go from there...
Homework Equations
The Attempt at a Solution
Homework Statement
Explain the difference between The principle of Mathematical Induction and The principle of Strong Induction. One is easier than the other. Why?
Homework Equations
The Attempt at a Solution
My book says that they are in fact identical. Proofs are almost the...
I'm trying to solve this problem from CH2 of spivak's calculus of which I am self-studying.
Homework Statement
Prove the following by mathematical induction:
1^3+...+n^3=(1+...+n)^2
Homework Equations
To prove by mathematical induction, you test whether P(1) is true and if P(k) is...
Homework Statement
Prove the Inequality for the indicated integer values of n.
n!>2^n,n\geq4
Homework Equations
The Attempt at a Solution
For n=4 the formula is true because
4!>2^4
Assume the n=k
k!>2^k
Now I need to prove the equation for k+1
I can multiply both sides by 2 and have...
Homework Statement
Use mathematical induction to prove the formula for every positive integer n.
\sum_{i=1}^{5}i^5=\frac{n^2(n+1)^2(2n^2+2n-1)}{12}
Homework Equations
The Attempt at a Solution
I know this will be true because the RHS is just your standard sums of powers of...
I cannot get my head around this at all.
Suppose that v is of type Vector of Integers and we know the following:-
1. v is sorted.
2. No two items in v are the same.
3. v.at(1) is 12.
For an integer n, where 1 <= n <= v.size(), let p(n) be the following proposition: p(n): v.at(n) => 11...
Homework Statement
All the terms of the arithmetic progression u1,u2,u3...,un are positive. Use mathematical induction to prove that, for n>= 2, n is an element of all positive integers,
[ 1/ (u1u2) ] + [ 1/ (u2u3) ] + [ 1/ (u3u4) ] + ... + [ 1/ (un-1un) ] = ( n - 1 ) / ( u1un)...
Homework Statement
prove:
0^2 + 1^2 + 2^2 + ... + n^2 = n(n+1)(2n+1)/6
Homework Equations
The Attempt at a Solution
I'm confused on how to prove this by induction. I'm not exactly sure what the goal of the rearrangement is after substituting (n+1). Any help is much...
Homework Statement
Prove that for all the natural numbers n that 2^n > n
Homework Equations
The Attempt at a Solution
Base Case is easy
Then the inductive step you have 2^k > k as the inductive hypothesis
show that p[k+1] holds
2^(k+1) > k+1
on the left side 2^(k+1) = 2^k * 2...
Hi
I'm a high school student. I gave a proof for the following theorem, but I was told by some professors that this is trivial and using natural induction twice for the rationals will do the same thing. What do you think? Is it just redundant?
Theorem:
Let P(r) be a statement about r...
Homework Statement
Prove that H1 +H2 +...+Hn = (n +1)(Hn-n)?
Homework Equations
Hn denotes the nth harmonic number.
The nth harmonic number is the sum of 1+1/2+...1/n,
which is n / n +1.
I'm not really sure if Hn = (1/ n) .
Prove by Mathematical Induction
Hn denotes the...
(a) Give a recursive definition of the set P of all non negative integers,
(b) formulate the applicable induction principle and
(c) then apply the induction principle to prove that 1/2^0+1/2^1+1/2^2...+1/2^i = 2-1/2^n for n>=0
I have solved parts a and b and stuck on c
(a) P is the...
Hello, In this problem I am trying to Determine the exact set of natural numbers n for which the inequality 2^n>=(n+1)^2 holds. (equation (1))
I have already dealt with the base case where n=6, (since the inequality does not hold for n<6), and so (1) holds for n=k, and I need to show that it...
Homework Statement
\forall n \in Z^+, \sum_{i=1}^n \frac{\sqrt{i+1}}{2i} > \frac{\sqrt{n}}{2}
Homework Equations
I have to prove the above via mathematical induction.
The Attempt at a Solution
I did the base case, n = 1 and found it true for the base case.
Then I assumed that...
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1. Homework Statement
Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.
Homework Statement
Let H be a ten- element set of potive integers ranged from 1 to 99. Prove that H has two disjoint subsets A and B so that the sum of the elements of A is equal to the sum of the elements of B.
Homework Statement
The Fibonacci numbers are defined by f(1)=1, f(2)=1 and for n>2, by f(n) = f(n-2) + f(n-1). Prove by mathematical induction that f(3n) is even for all natural numbers n.
Proof:
Base case: (n=1)
f(3) = f(2)+f(1) =1+1 =2 is even
Induction hypothesis: (suppose the...
This is not homework, per se, but I have recently started reading Courant and Robbins' What is Mathematics?
Homework Statement
Prove by mathematical induction.
(1+q)(1+q2)(1+q4) ... (1+q2n) = (1-q2n+1) / (1-q)
Homework Equations
Only the problem itself.The Attempt at a Solution
Using the...
Homework Statement
Prove by matematical induction that (2^(n+1)+9(13^n)) divides by by 11 for all positive intergers
Homework Equations
The Attempt at a Solution
I really have no idea where to start...
Homework Statement
Is mathematical induction a deductive or inductive argument?
Would appreciate the help. Thanks.
Jeremy
Homework Equations
The Attempt at a Solution
Its name suggests that the process is inductive, yet I know all of mathematics depends on deductive reasoning...
Hi,
I need some help with mathematical induction
The question is as follows:
prove that \sum_{i=0}^n (i-3) \geq \frac{n^2}{4}
I have shown that it holds for the base step where n=12
\frac{144}{4} = 36
and the sum of all the i's up to 12 -3,-2,-1,0,+1,+2,+3,+4,+5,+6,+7,+8,+9 = 39...
I was given the problem: For n \geq 1, 2 + 2^{2} + 2^{3} + 2^{4} + ... + 2^{n} = 2^{n+1} – 2.
I did the induction on it and got 2^{k+2}-2. I know this is the right answer but I don't know WHY. Could anyone explain it to me?
The equation is: [i(i+1)=n(n+1)(n+2)/3] whereas i=1
so the beginning process would be 2+6+12+20...+n(n+1)=n(n+1)(n+2)/3
after the equation is proven for n=1 [(1(1+1)=1(1+1)(1+2)/3] then we must prove for n=n+1
thats where i begin to stop understanding.
So...
I know this sounds kind of like a basic question, but why does the method of mathematical indcution work to prove things like sequences and such? All the other proof methods I have learned have made sense to me, and I can prove using logical truth tables or axioms, but I don't really get how...
Homework Statement
prove, using mathematical induction, that the next equation holds for all positive t.
\sum_{k=0}^n \dbinom{k+t}{k} = \dbinom{t+n+1}{n}
Homework Equations
\dbinom{n}{k} = {{n!} \over {k!(n-k)!}The Attempt at a Solution
checked that the base is correct (for t=0, and even for...
While learning mathematical induction,an idea occurred to me.
Mathematical induction is done by proving that the first statement in the infinite sequence of statements is true, and then proving that if anyone statement in the infinite sequence of statements is true, then so is the next one...