# Tagged Questions

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### Prove by induction about line set

A set of straight lines in the plane is said to be in general position if no two lines are parallel and no three lines intersect at a common point. Consider $n\geq3$ lines in general position in the ...
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### Prove via induction this recursively defined sequence

Let $P(n) = 2P(n-1) + n, P(1) = 3.$ Use induction to show that $$P(n) = 3(2^n) - n - 2$$ Highly verbose solutions are greatly appreciated.
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### Prove that $3+ 3\cdot5+…+3\cdot5^n = \frac{3(5^{n+1}-1)}{4}$ for all nonnegative integers.

I have been stuck on this one for a while. Supposed to use induction to prove that $3+ 3\cdot5+...+3\cdot5^n = \Large\frac{3(5^{n+1}-1)}{4}$ for all nonegative integers. I don't know if I'm taking ...
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### Prove by Induction (Geometric Progression)

Prove by induction that for any real number $q≠1$ and any $n\in \mathbb N$ we have $\sum_{i=0}^n q^i=\frac{q^{n+1}-1}{q-1}$
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### Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$

Prove that, for any positive integer n: $(a + b)^{n} \leq 2^{n-1}(a^{n}+b^{n})$ I tried induction theorem, when $n = 1$ it is obviously right. But, say $n=k$, It does not make sense since I cannot ...
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### How to prove this Mathematical Induction problem?

We got $n \geq 3$ lines drawn on a surface with conditions below: No two lines are parallel. No three lines make a conjunction in a specific point. Prove that one of the areas created by these ...
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### How to prove this claim using Mathematical Induction?

We have $n$ points on a surface and for each $3$ points, we are able to put them into a circle with radius of unit length. Prove that all of these points are on circle with radius of unit length. My ...
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### How to prove this equation using Mathematical Induction?

I was trying to prove this. I tried somehow but didn't get any idea. I think we can prove this using induction. I'd really appreciate it if you could help me. ...
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### Question over regular induction: Let $P(n)$ be the statement that $n$-cent postage can be formed using just 4-cent and 7-cent stamps

Prove $P(n)$ is true for $n \geq 18$ using regular induction. I know how to do this problem using strong induction but don't know how to proceed using regular induction. I know the first step is ...
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### Mathematical Induction

I've gotten to the final step and believe my problem lies within my algebra. Prove the following: $1 \times 3 + 2 \times 4 + 3 \times 5 + ... + N(N+2) = \frac{N(N+1)(2N+7)}6$ Here is my show that ...
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### Question over induction, suppose $P(n)$ is true for all positive integers $n$ that is a power of 2.

Suppose, that $P(k+1) \Rightarrow P(k)$ for all positive integers $k$. How would I prove $P(n)$ is true? I am getting confused since this is going the 'other way'. Usually $P(k)\Rightarrow P(k+1)$
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### Question about induction, if $P(k)$ implies $P(k+3)$ [closed]

Suppose $P(1)$ and $P(2)$ are true. For what values of $n$, is $P(n)$ true for if for every positive integer $k$,if $P(k)$ is true then $P(k+3)$ is true?
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### Prove divisibility by using induction

Prove that for integers $n > 0$, $n^3 + 5n$ is divisible by $6$. Here is what I have done: Base Step: $n=1$, $1^3+5(1)=6$ Inductive Step: $p(k)=k^3 + 5k =6m$, $m$ is some integer ...
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### Prove that $n^2 < 2^n$ for all $n \geq 6$

My approach to solving this: By induction. (1) $S(n) = (n^2 < 2^n)$ for all $n \geq 6$, $n \in \mathbb N$. (2) Base Case: $n = 6$ $$6^2 < 2^6$$ $$36 < 64$$ So the statement is true for ...
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### Strong Induction: Finding the Inductive Hypothesis

Consider this claim: Every positive integer greater than 29 can be written as a sum of a non-negative multiple of 8 and a non-negative multiple of 5. Assume you are in the inductive step and trying ...
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### Induction of $A_i$ [duplicate]

The base case $n=1$: $B\cup\left(\bigcap_{i=1}^1A_i\right)=B\cup A_1$ and $\bigcap_{i=1}^1(B\cup A_i)=B\cup A_1$. Now, suppose inductively that ...
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### Discrete Math: Ways to Prove Induction

The point of mathematical induction is to prove $\forall x\geq b[P(x)]$ by instead proving $P(b)\wedge \forall x\geq b[P(x)\rightarrow P(x+1)]$ ($b$ is often, but not always, $0$ or $1$). However, ...
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### Proof by induction that $B\cup (\bigcap_{i=1}^n A_i)=\bigcap_{i=1}^n (B\cup A_i)$

$B\cup (\bigcap_{i=1}^n A_i)=\bigcap_{i=1}^n (B\cup A_i)$ I was able to prove this without using induction, however I am supposed to prove it using induction. How should I go about doing so?
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### Is it possible to play the Tower of Hanoi with fewer than $2^n-1$ moves?

The Tower of Hanoi game consists of three identical upright pegs and n rings all of different diameters that can be stacked over any of the pegs. Initially, all of the rings are stacked around one of ...
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### Mathematical Induction Recursion

Consider the recursion given by $f(n) = 2f(n−1)− f(n−2)+6$ for $n ≥ 2$ with $f (0) = 2$ and $f (1) = 4.$ Use mathematical induction to prove that $f (n) = 3n^2 −n+2$ for all integers $n ≥ 0.$
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### Counting tilings of a $2\times n$ board

Let $n=>1$ be an integer and consider a $2*n$ board $B_n$ consisting of $2n$ cells,each one having sides of length one. This picture shows $B_{13}$: For $n=>1$, let $a_n$ be the number of ...
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### Proving some sequence of integers by induction

Say I have a sequence like: $0,1,2,0,1,2,0,1,2,\dots$ in other words $1=0$, next $2=1$, third $3=2$ etc. and a formula that I believe works for my sequence. How would I prove that the sequence works ...
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### Proving by induction inequalities that lack the variable on the right side.

Doing proof by induction exercises with inequalities, I got stuck on one that is a bit different from the others. There is no $n$ term on the rightmost part of the inequality: Prove that the ...
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### Proving by induction that $1+\frac{1}{2}+\frac{1}{3}+…+\frac{1}{n}\le\frac{n}{2}+1$ holds for all $n \ge 1$

While looking at some examples of proof by induction related to inequalities, I had this one that I didn't quite get: Prove by induction that the following holds for all $n \ge 1$: ...
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### What is the purpose of the first test in an inductive proof?

Learning about proof by induction, I take it that the first step is always something like "test if the proposition holds for $n = \textrm{[the minimum value]}$" Like this: Prove that ...