# Tagged Questions

For questions about mathematical induction, a method of mathematical proof. Mathematical induction generally proceeds by proving a statement for some integer, called the *base case*, and then proving that if it holds for one integer then it holds for the next integer. This tag is primarily meant ...

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### Differentiate a geometric sum and show that it is less than an equation

The question: a) Differentiate both sides of the geometric series with respect to $r$: $~~\displaystyle\sum_{i=0}^nr^i=\frac{1-r^{n+1}}{1-r}$ b) Use the result in part (a) to show that ...
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### Proving $\frac{n}{n+1} < \frac{n+1}{n+2}$ by induction?

I have the inequality $\frac{n}{n+1} < \frac{n+1}{n+2}$ I'm not sure how to go about proving it. I've started by testing with n = 1, which results in $\frac{1}{2} < \frac{2}{3}$ which is true ...
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### Proof that $\lfloor x + x\sqrt3 \rfloor = x + \lfloor x\sqrt3 \rfloor$ Where $x \in N$

I'm not sure if this is true or not. I was going to prove it via induction. Here is what I have so far. Base Case (x=0): $\lfloor 0 + 0\rfloor = 0 = 0 + \lfloor 0 \rfloor$ Induction Hypothesis (IH): ...
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### Any collection of n coins can be obtained using a combination of 3¢ and 5¢ coins where n ≥ 14

I am trying to prove this statement with strong induction, but I'm a little lost on the inductive step. Proposition: Let P(n) be the sentence ‘any collection of n coins can be obtained using a ...
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### Structural induction on List

In preparation for an exam, I've come upon the following problem. Given the constructors : ...
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### Finding an expression for the sum of n tems of the series $1^2 + 2^2 + 3^2 + … + n^2$ [duplicate]

Possible Duplicate: why is $\sum\limits_{k=1}^{n} k^m$ a polynomial with degree $m+1$ in $n$ I know that if you have a non-arithmetic or geometric progression, you can find a sum $S$ of a ...
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### Require assistance proving $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$

Theorem: $n≥2 \Longrightarrow \frac{n!}{n^n} ≤ \frac{1}{2}^{\lfloor \frac{n}{2}\rfloor}$ Attempted Solution: We use induction. Additionally, we prove the stronger inequality omitting the floor ...
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### Proof by induction: $(a+b)^n=a^n+na^{n-1}b+\frac{n(n-1)}{2!}a^{n-2}b^2+…+nab^{n-1}+b^n$

I'm studying in preparation for a Mathematical Analysis I examination and I'm solving past exam exercises. If it's any indicator of difficulty, the exercise is Exercise 1 of 4, part $a$ and graded for ...
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### Prove $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0

The statement I'm trying to prove is: $n^3 + 7n + 3$ is divisible by 3 for all integers n ≥ 0 I eventually need to prove $(k + 1)^3 + 7(k + 1) + 3$ is divisible by 3. I don't really understand ...
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### Prove that $f(n) = 3n^5 + 5n^3 + 7n$ is divisible by 15 for every integer $n$

So far I have only been able to complete the base case for which I got the following: $$f(n) = 3n^5 + 5n^3 + 7n$$ $$f(n) = 3(1)^5 = 5(1)^3 + 7(1)$$ $$f(n) = 3 + 5 + 7$$ $$15/15 = 1$$ From here ...
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### 'Mathematical Induction'

Use mathematical induction to prove that $4^n -3^n + 1 = 7a_{n-1} – 12a_{n-2} + 6$ with $n \ge 3$ with the initial condition $a_1 = 2$ and $a_2 = 8$ . Given that $a_n = 4^n -3^n + 1$. I am confused ...
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### Induction proof: $(x_{1}+x_{2}+…+x_{n})^2\leqslant n(x_{1}^2+x_{2}^2+…+x_{n}^2)$ [duplicate]

I have a problem proving this inequality but I can't get anywhere after the inductive step. $$(x_{1}+x_{2}+...+x_{n})^2\leqslant n(x_{1}^2+x_{2}^2+...+x_{n}^2)$$ Maybe some hints would help.
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