# sequence

1. ### Finding a formula for a sequence

Can anyone please help me find a formula for this sequence: 4, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 10, 10, 10, 11, 11, 11, 11, 12, 12, 12, 12, 12... It's based on fewest number of 2-blocks required to surround an increasing number of 1-blocks. I've tried the method where you find differences...

I tried asking this question in the "Algebra" section, but it's been almost a week and I've got zero answers. Please forgive me if this is not the right place to ask, but I'm not sure what else to do. Question: How can I calculate the formula behind the following sequence:when $x = 100$, then...
3. ### What is wrong in this GM >= HM ?

Find the minimum value of (1+4 cosec x)(1 + 8sec x) for 0< x < pi/2. I tried doing by GM>= HM \text{Let }\sqrt{a} = 1+ 4 \csc x \text{ and } \sqrt{b} = 1+8 \sec x \\GM \geq HM \\ \\ \sqrt{ab} \geq \frac{2ab}{a+b} \\ \\ \text{Squaring on both sides,}\\ ab \geq \frac{ab \cdot 4ab}{(a+b)^2}\\...
4. ### sequences

Look at the sequences below: a) 198, 205, 212, 219, 226,... b) 196, 205, 214, 223, 232,... What is the first number after 205 that will appear in both sequences? (Tip: Try to work this out without​ writing out every term in the sequence!)
5. ### Help with arithmetic sequence

In an arithmetic sequence, the seventh term is 3 and the sixteenth term is 9. use arithmetic means to determine the common difference and the first term of the sequence Thanks for the help!
6. ### Sequence Convergence Proof

Hi, I've been reading a book in order to prepare for a heavy on the proofs calculus course in the fall and have run into some trouble with the following proof Prove that the sequence $(a_n)$ given by $a_n = 3 - \frac{4}{n}$ $\forall n \in \mathbb{N}$ converges to 3. The final lines of the proof...
7. ### Analysis: show xn is a null sequence

Prove that x_{n+1}-1/2x_n\to 0 \Rightarrow x_n\to 0. My approach: Let \varepsilon>0. Then there exists an N\in \Bbb N s.t. \forall n\geq N, |x_{n+1}-1/2x_n|<\varepsilon. It follows that -\varepsilon\leq -\varepsilon+1/2x_n<x_{n+1}<\varepsilon \leq \varepsilon+1/2x_n. Then for any n\geq N, we...
8. ### Sequence

Pls help me to answer this common difference sequence ...problem given a1=3 a15=31 find d ?
9. ### Two general sequence summation

As shown in Chinese wikipedia "Summation" For \sum_{k=1}^n p(k) \begin{align*}\sum_{k=1}^n p(k)=\begin{pmatrix}C_n^1 & C_n^2 & \cdots & C_n^{m+1}\end{pmatrix}\begin{pmatrix}C_0^0 & 0 & \cdots & 0\\-C_1^0 & C_1^1 & \cdots & 0\\\vdots & \vdots & \ddots & \vdots\\(-1)^mC_m^0 &...
10. ### Proving a Sequence's Convergence

Hello, can you please check my work? Prove sin(n^2)/n^(1/3) converges to 0 Proof for any e > 0 abs(sin(n^2)/n^(1/3)) < e sin(n^2)/n^(1/3) < e Since the range of sin(n^2) is [-1,1], sin(n^2)/n^(1/3) < or = 1/n^(1/3) so let 1/n^(1/3) < e. So 1/n < e^3 let N < or = n Then, sin(n^2)^3/n < or =...
11. ### Finding the Limit of a sequence

Hello, I have encountered a question where I need to show the limit exists and then find it when I am given S1=1 Sn+1=1/3(sn+1) Any help would be thoroughly appreciated
12. ### Number Sequence Game...

If you like number sequence puzzle. Check this out https://play.google.com/store/apps/details?id=com.elyeproj.numbersequence. Have fun. We could discuss some of the difficult puzzle here.
13. ### Convergence of a sequence?

I have this sequence: And I have to see if it converges. I tried starting with proving this (based on the known convergence criterion for sequences): But I can't simplify that fraction ... I've been stuck for more than an hour busting my head. Excuse my english. Thanks for your time. EDIT...
14. ### Why does this sequence converge? n * sin ( 1/n )

Hello, The back of my textbook says that this sequence converges and that its limit is 1. Can you explain this to me? I don't understand how to arrive at that answer. {an} = n*sin(1/n) What theorem, law, etc. should I apply?
15. ### Monotonic Sequence

{an} is a monotonic, increasing sequence. Show {an} has a convergent subsequence iff {an} is bounded. The obvious route here is by way of contradiction. I can assume {an} is unbounded. This means there is no L st. L <= ai for all ai and there is no U s.t. U>= ai for all ai. However, I can...
16. ### Finding a formula for an exponential sequence

The sequence is defined as U_{n}=3+2U_{n-1} where U_{1}=3 n=... 1 2 3 4 5 U_{n} 3 9 21 45 93 2^n 2 4 8 16 32 I know that a 2^n has to be involved somewhere as the sequence is generated by doubling the previous term, but I'm not sure what to do next..
17. ### Recursive sequence

$\text{I don't know how to find the general formula for the sequence: }\\ (x_n):x_{n+1}=\frac{ax_n+b}{cx_n+d},\text{ with } x_1,a,b,c,d\in\mathbb{R} \text{ given.}\\ \text{I know there are some formulas but a couldn't find them. Please help!}$
18. ### Geometric Sequence Problem

We are working with the explicit formulas for arithmetic AND geometric sequences. I get for the most part, but it is the world problems that stump me a little. Here's one that confused me that I could use help on: John buys a computer for 1200 dollars. He predicts that the value of the...