# Tagged Questions

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### Proof by induction (exponents)

Use proof by induction and show that the formula holds for all positive integers:$$1+3+3^2+\dots+3^{n-1}=\frac{3^n -1}2$$ The confusing step in my opinion is the first expression: $3^{n-1}$, when ...
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### The number $(3+\sqrt{5})^n+(3-\sqrt{5})^n$ is an integer

Prove by induction that this number is an integer: $$u_n=(3+\sqrt{5})^n+(3-\sqrt{5})^n$$ Progress I assumed that it holds for $n$ and I tried to do it for $n+1$ but the algebra gets quite messy and ...
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### How to simplify the formula for $n$th Fibonacci number when $n=2$?

When n is equal to 2 how do I simplify when the $n=2$ is put into the equation below (by the way I have to prove this formula by induction that when n= any number it will equal that number) ...
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### Clarification regarding the Josephus problem in Concrete Mathematics (Knuth, et al)

In page 9 of Concrete Mathematics, regarding the Josephus Problem, they state that "each person's number has been doubled then decreased by 1". $J(2n) = 2J(n) - 1$, for $n \ge 1$ I don't quite ...
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### Mathematical Induction Problem with Fraction

$$(3n-2)^2=\frac{n(6n^2-3n-1)}{2}$$ I can't seem to solve it out to the point where I can prove it right or wrong. I always hit some sort of roadblock where I don't have enough info to prove it ...
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### Spivak Chapter 2 Question 1 (i)

I don't understand Spivak's proof by induction of this exercise: Prove by induction $$1^2 + \ldots + n^2 = {n(n+1)(2n+1))\over 6}$$ It's true for $n = 1$ Then the proof continues adding $(k+1)^2$ ...
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### Analysis, prove a period by induction

Given that $F(x) = F(x+T)$ is $T$-periodic, prove by induction that $F(x) = F(x+nT)$ for all $n \in \mathbb N$. Would appreciate some help with this... one of my finals practice questions. Thanks.
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### Finding a proof to the 'squares' problem

I am trying to find a proof for the general case of the solution to the 'Squares' Problem. This is what I have managed to figure out: If n is the number of squares in the top row, then the number ...
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### Use induction to show that $3^n >n^3$ for $n≥4$

Use induction to show that $3^n >n^3$ for $n≥4$. (Note that you have to start at $n=4$ as the result isn't true for $n=3$ !) I am very new to using induction, but as I understand it I have ...
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### Prove by mathematical induction for every natural number n. $5+25+125+\cdots+5^n=5/4(5^n-1)$

There's one thing I don't understand. In the work shown for this problem in the image below, why is it adding $5^{k+1}$ to both sides? http://imgur.com/d369K5Y (Part 1) http://imgur.com/X9Q6aTi ...
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Prove: $\displaystyle 1+2+\ldots+n=\frac{n(n+1)}{2}$. Proof When $n=1,1=\displaystyle \frac{1(1+1)}{2}$,equality holds. Suppose when $n=k$, we have $1+2+\dots+k=\frac{k(k+1)}{2}$ When $n = k + ... 1answer 81 views ### Why is this summation formula wrong? This is the alternate form of the summation formula: $$\sum^{n}_{k=0} a(c)^k = \frac{ac^{n+1} - a}{c - 1}$$ so why is this wrong? $$\sum^{n}_{k=0} (-\frac{1}{2})^k = \frac{(-\frac{1}{2})^{n+1} - ... 2answers 89 views ### Two exercises on mathematical induction Studying for a test and can't work out how to do two questions on the sample test. (1) Suppose a sequence of numbers a_1, a_2, \dots is defined recursively by:$$a_1 = 1\qquad\text{and}\qquad ... 2answers 126 views ### I need help with proofs using mathematical Induction I need help with this problem:$2+7+12+17+...+(5n-3)=(\frac{n}{2})(5n-1)$1answer 69 views ### Prove by induction$ \sum^n_{i=1}(i-1/2) = n^2/2 \$
This is a question from a test that I wrote and I'm wondering how do you solve it. Prove by induction that $$\sum^n_{i=1}(i-1/2) = \frac{n^2}{2}$$ *Provide a Base Case, Inductive Hypothesis, and ...