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Questions tagged [faq]

This is meant for questions which are generalized forms of questions which get asked frequently. See tag details for more information.

When you add a question marked faq, please also update the list of questions: List of Generalizations of Common Questions

The question which prompted this: Coping with *abstract* duplicate questions.

Note: Even though one might argue that tagging a question as faq should be enough, and there is no need to update the above list, updating the above list will serve to bring this policy back to attention and help raise awareness periodically.

112 questions
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Given $\langle f_{n}\rangle$ such that $f_{n}=\frac{f_{n-1}+f_{n-2}}{2}$ $\forall n\gt2$,to prove it converges to $\frac{f_1+2f_2}{3}$

If $\langle f_{n}\rangle$ be a sequence of positive numbers such that $$f_{n}=\frac{f_{n-1}+f_{n-2}}{2}$$ $\forall n\gt2$ ,then show that $\lt f_{n}\gt$ converges to $$\frac{f_1+2f_2}{3}$$ Replacing $n$ by $3,4,5,....,n-1$,we…
PiGamma
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2 answers

Proving a general rule which states where a recursive series converges

The recursive formula is $t_n=\frac {t_{n-1}+t_{n-2}}2$ Changing $t_1$ and $t_2$ changes the number where the sequence converges as $n \to \infty$. With the help of everyone at StackExchange, I found that the relationship between $t_1$ and $t_2$ and…
Jane
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1 answer

Sterling Numbers of The Second Kind With Limitations Placed on Boxes/Parts

I know there are similar problems already on the board. However, none of the previously stated questions contain problems where limitations are placed on the BOXES. Thus, seeing that I am struggling with finding answers to these limitation…
KING
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1 answer

Volume of Intersection of cylinders (different radii)

I want to derive a formula for the area of the intersection of two rigth cylinders with different radii. To get an idea I attached a sketch. My idea is to determine the borders of $x$ and $z$ in dependence of $y$. The borders of $x$ are:…
fmeyer
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4 answers

Show that for all $n ≥ 2$ it is true: $1^3+2^3+\cdots+(n-1)^3<\dfrac{n^4}{4}$

How can I prove that? $1^3+2^3+\cdots+(n-1)^3<\frac{n^4}{4}$
Rubén Darío Guarnizo Navarro
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3 answers

Limit of a mean sequence

The recursive formula is $$t_n=\frac {t_{n-1}+t_{n-2}}2$$ as $n$ approaches infinity the mean sequence converges at a certain number. Changing $t_1$ and $t_2$ changes the number where the sequence converges. Find a relationship between $t_1, t_2$…
Jane
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Let $0 < a_1 < a_2$. Consider the sequence $a_n$ defined by $a_{n+1}$ = $(a_n + a_{n−1})/2$. Show that lim $a_n$ = $(a_1 + 2a_2)/3$

Let $0 < a_1 < a_2$. Consider the sequence $a_n$ defined by $a_{n+1} = \frac{a_n + a_{n−1}}{2}$. Show that $\lim _{n \to \infty}a_n = \frac{a_1 + 2a_2}{3}$. I have tried substituting multiple terms, like $a_3 = \frac{a_1+a_2}{2}$, then $a_4…
lk1313
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