1
vote
1answer
8 views

Insert Means in an Arithmetic Sequence (that contains logarithms)

So the question is: You have an Arithmetic Sequence. Log 2 and Log 1024 are two terms in the sequence Find 8 arithmetic means between them.
0
votes
2answers
30 views

Finding the fifth term of this arithmetic progression problem

If the sum of the first 10 term is equal to $-240$ and the 7th term is $48$. How to find the 5th term?
0
votes
1answer
19 views

Derive a formula to get the particular value from table

I have a table of points earned given the final game score. ...
1
vote
2answers
43 views

Bounding $\sum_{n=n_1}^\infty x^n (n+1)^2$

I need to upperbound the sum $$\sum_{n=n_1}^\infty x^n (n+1)^2$$ where $0<x<1$ is a parameter. I know it can be done starting from $$\sum_{n=n_1}^\infty x^n (n+1)^2\le \sum_{n=0}^\infty x^n ...
1
vote
4answers
101 views

Series question

I was trying this question: Consider the infinite series $$\frac{1}{1!} +\frac{4}{2!}+\frac{7}{3!}+\frac{10}{4!}+...$$ If the series continues with the same pattern, find the an expression for the ...
2
votes
5answers
97 views

How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing?

How to prove/show that the sequence $a_n=\frac{1}{\sqrt{n^2+1}+n}$ is decreasing? My idea: $n^2<(n+1)^2 /+1$ $n^2+1<(n+1)^2+1/ \sqrt{}$ $\sqrt{n^2+1}<\sqrt{(n+1)^2+1}/+n$ ...
1
vote
2answers
38 views

Simplifying ratio test with exponents $k+1$

Question: Find the interval and radius of convergence. $$\sum_{k=1}^\infty\frac{(x-1)^k(k^k)}{(k+1)^k} .$$ I applied the ratio test. ...
0
votes
3answers
79 views

Finding the $n$th term for the sequence $1, \frac{1}{2}, 3, \frac{1}{4}, 5, \frac{1}{6}, \dots$

I have tried using a negative exponent. I need one statement not two, the pattern is $$1, \; \frac{1}{2}, \; 3, \; \frac{1}{4}, \; 5, \; \frac{1}{6}, \dots$$
1
vote
1answer
24 views

How many points determine a two variable polynomial of degree n+k?

I am working with sequences and it would be extraordinarily useful to have a two variable version of the following: A degree-$n$ polynomial is uniquely characterized by its values at any $n+1$ ...
3
votes
1answer
149 views

Sum this series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\ldots$ upto $n$ terms

Sum this series: $$\dfrac{1}{1+1^2+1^4}+\dfrac{2}{1+2^2+2^4}+\ldots$$ upto $n$ terms. My approach: $$(1-n^6)=(1-n^2)(1+n^2+n^4)\implies \dfrac{n}{1+n^2+n^4}=\dfrac{n(1-n^2)}{1-n^6}$$ So, the ...
1
vote
2answers
64 views

Periodic sequence [duplicate]

$(x_n)_n$ is a sequence defined by the relation: $x_{n+2}=|x_{n+1}-x_{n-1}|$ for $n\geq1$ and $x_0,x_1,x_2$ are non-negative integers, not all three equal 0. I think this sequence is periodic, so here ...
5
votes
2answers
107 views

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$

Prove that $\sqrt{8}=1+\dfrac34+\dfrac{3\cdot5}{4\cdot8}+\dfrac{3\cdot5\cdot7}{4\cdot8\cdot12}+\ldots$ My work: $\sqrt8=\bigg(1-\dfrac12\bigg)^{-\frac32}$ Now, I suppose there is some "binomial ...
0
votes
2answers
28 views

What's the maximum deviation from loan amortization

Suppose you have a loan with principle P and fixed interest rate i compounded daily. Suppose you make fixed payments every month, but not on the same day. The only constraint is that you make every ...
4
votes
3answers
181 views

Closed-Form Solution to Infinite Sum

Does the convergent infinite sum $$ \sum_{n=0}^{\infty} \frac{1}{2^n + 1} $$ have a closed form solution? Quickly coding this up, the decimal approximation appears to be $1.26449978\ldots$
1
vote
0answers
63 views

Is there a general product formula for $\sum\limits_{k=1}^{n} k^p$

I'm familiar with Faulhaber's formula to express this sum as a much simpler one, but it appears that for any $p$ there's a product formula in $n$ for the sum e.g.: $$\begin{align} & ...
1
vote
2answers
70 views

proof by maths induction

not sure how to prove this: for all positive intergers prove: \begin{equation} 1+2(2)+3(2^2)+...+n(2^{n-1})=(n-1)(2^n)+1 \end{equation} heres my try: prove $n=1$ : \begin{equation} 1=1 ...
1
vote
1answer
35 views

Complex numbers summation

Sum the series $$\sum_{r=0}^n {{n}\choose{r}} \sin(\alpha +r\beta)$$ I've been using the C+jS method where C is the cosine series and S is the sine series and forming a result from there but have not ...
2
votes
2answers
38 views

a problem on arithmetic progression,it is a very confusing sum

In an arithmetic progression,the sum of five terms is equal to 1\4 of the next five terms,prove that the 20th term is -112?
3
votes
3answers
96 views

Sequence is periodic $x_{n+2}=|x_{n+1}-x_{n-1}|$

How to show that the sequence $$x_n, n \geq 0, x_{n+2}=|x_{n+1}-x_{n-1}|, n \geq 1$$ with $x_0, x_1, x_2$ positive integers, not all null, is periodic? I tried to pick up the square but obtained ...
1
vote
1answer
40 views

Sum of G.P. terms to infinity when r $ \lt $ 1

The sum of n g.p. terms with first term a and common ratio r is given by $$ S_{n} = a\cdot\frac{1 - r^n}{1 -r} provided \ r \not= 1 $$ But I'm confused as to what happen when r $ \lt 1$. My module ...
1
vote
2answers
30 views

Polynomials through successive differences

Let $h_0:\Bbb{N}\rightarrow\Bbb{N}$ be any function. Define recursively, for $m>0$, $$h_{r+1}(m)=h_r(m)-h_r(m-1).$$ Suppose that for some $k>0$ we have $h_k(m)\equiv d$ constant. Is this ...
0
votes
1answer
91 views

Arithmetic progression question

In an arithmetic progression there are $2n$ arguments. The sum of last $n$ arguments is three times greater than that of the first n arguments. It is also known that the last argument in the series is ...
1
vote
3answers
71 views

Is this geometric sequence convergent or divergent?

2, -6, 18, -54,... Is this geometric sequence convergent or divergent? I know that convergent sequences have terms that are approaching a constant, but how do I find out if that is the case? Thanks! ...
0
votes
1answer
23 views

Bounding the sum of “almost” factorials

I am analyzing the complexity of an algorithm and the result is the sum of n products. Product 1 is the factorial. Product 2 is the factorial divided by 2. Product 3 is the factorial divided by 3 etc. ...
0
votes
1answer
51 views

arithmetic and geometric progression

$a,b,c$ ,are consecutive arguments of a given arithmetic progression. $a^2, b^2, c^2$, are consecutive arguments of a geometric progression. With this information given, find the $q$ for the ...
7
votes
3answers
256 views

Sum of a Hyper-geometric series. (NBHM 2011)

How to find the sum of the following series $$\frac{1}{5} - \frac{1\cdot 4}{5\cdot 10} + \frac{1\cdot 4\cdot 7}{5\cdot 10\cdot 15} - \dots\,.?$$ I have no idea. I have written the general ...
0
votes
1answer
52 views

Determine positive integers

If $n\geq3$ is a positive integer, determine $n+1$ positive integers With the property that the sum of all $n$ integers from the $n+1$ numbers build the set: ...
3
votes
5answers
441 views

Condition for a common root in two given quadratic equations

If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in: Arithmetic Progression ...
1
vote
1answer
86 views

Find $x$ in $\large \,\, A^{A^{{{A^.}^.}^.}}= \,\,(\sqrt [x\cdot x\cdot x\cdot …] x)^{ (((1\cdot x+1)x +1)x +1)x+1…} $

If $\large \,\, A^{A^{{{A^.}^.}^.}}= \,\,(\sqrt [x\cdot x\cdot x\cdot ...] x)^{ (((1\cdot x+1)x +1)x +1)x+1...} $, and $\large \,\,A = (\sqrt[3]{3\sqrt 3 })^{\frac{\sqrt 3}{3}} $, find $x$ I have ...
1
vote
1answer
45 views

Finding the value of a logarithmic expression involving an infinite GP

Find the value of $(0.16)^{\displaystyle\log_{2.5}(\frac13+\frac1{3^2}+\frac1{3^3}+\cdots)}$. I could solve the series. It gave $$(0.16)^{\log_{2.5}0.5}$$ Unable to proceed from here.
0
votes
2answers
45 views

(Geometric) Sum of $100-50-25-(25/2)+\ldots+ (25/16)$

Determine the sum for this geometric series: $100-50+25-(25/2)+\ldots+ (25/16)$ I found $7$ to be the number of terms in this series, and the sum of the series to be $67.1875.$, but, the ...
0
votes
2answers
85 views

Common difference between terms of the arithmetic progression

In an finite arithmetic progression: $S_5=55$, sum of the last five terms is 215 and total sum is $S_n=351$ . What is common difference between terms of the arithmetic progression? Why?
1
vote
2answers
45 views

Could someone explain the algebra in this sequences problem?

$$ \frac{(n^2+n)-n^2}{\sqrt{n^2+n}+n} = \frac{n}{n\left(\sqrt{1+\frac{1}{n}}+1\right)} $$ The advantage here is that the n's cancel, but I don't see why the argument of the sqrt() function ends up ...
1
vote
2answers
229 views

Help find sum to infinity of a series - odd numbers with a common ratio

I am trying to derive the formula for the variance of a geometric distribution and am stuck at the following problem: I need to find the sum to infinity for the following series: ...
2
votes
2answers
43 views

Finding $x$. The summation of the floor of the equation.

I would appreciate if somebody could help me with the following problem Q:Finding $x$. The summation of the floor of the equation. $$\sum_{i=1}^{2013}\left\lfloor\frac{x}{i!}\right\rfloor=1001$$
0
votes
1answer
32 views

Math behind Keynesian Expenditure Multiplier

Take a look at this page: http://wiki.ubc.ca/Keynesian_Multiplier Why can you find out the sum of the geometric series just by dividing the mps by 1?
1
vote
1answer
186 views

Simplification of $\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}}$

I'm having trouble understanding how this expression: $$\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\dots}}}} \cdot ...
4
votes
2answers
49 views

Rational sequence with $a_{n+1}=2a_n^2-1$

Suppose we start with a rational number $a_0$, and define $a_{n+1}=2a_n^2-1$ for $n\geq 0$. For what $a_0$ will it be the case that $a_i=a_j$ for some $i\neq j$? We can start with something like ...
0
votes
1answer
60 views

Sequence Puzzle

If $a_{0}$= 1 , $a_{1}=1,$ $a_{n}=a_{n-1}a_{n-2}+1$for $n>1$ then A) $a_{465}$ is odd and $a_{466}$ is even B) $a_{465}$ is odd and $a_{466}$ is odd C) $a_{465}$ is even and $a_{466}$ is even ...
0
votes
1answer
71 views

Solving this recursive function $f(x)=f(x-k)+f(x/k)$.

How to solve or simplify the following recursive function? $f(x)$ is defined only for whole numbers as follows: $$f(x)=\begin{cases} 1 & \mbox{if } x<k; \\ f(x-k)+f(x/k) ...
0
votes
1answer
50 views

Geometric sequences interest question

Winston invests a sum of money at 6% per annum. How many years does it take him to double his money? I let the initial sum of money be $£a$. Then at the end of the first year, he has $£1.06a$ since ...
5
votes
3answers
184 views

Error in finding sum of $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$

To find sum of the series $1\cdot 2+3\cdot 4+ \cdots \text{to}\space n\space \text{terms}$ My approach, Let S=$1^2+2^2+3^2 + \cdots +n^2$ If $n$ is even S=$(1-2)^2+(3-4)^2+ \cdots +[(n-1)-n]^2+2(1 ...
3
votes
2answers
46 views

Solve $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+…+x^{2008}\right)=2010x^{2009}$

Solve for $x$ $\left(x^{2010}+1\right)\left(1+x^2+x^4+x^6+.......+x^{2008}\right)=2010x^{2009}$ solution should be by hand
2
votes
5answers
158 views

What is the $46^{th}$ term in the sequence $27,-20,13,-6,-1,8,-15$

I am doing some practice algebra from brilliant.org and I am having some trouble with the following question: What is the $46^{th}$ term in the sequence $27,-20,13,-6,-1,8,-15$ I tried ...
0
votes
1answer
42 views

Solution of a series equation containing $\pi$

I have to find the coefficients $a_k$ for which the following equation is satisfied: $$S(N)=\sum_{k=1}^Na_k\frac{\pi^{k+3}}{k+3}=-2\pi+\epsilon(N)$$ where $\epsilon(N)$ is an error depending on $N$. I ...
1
vote
2answers
57 views

Prove geometric sequence question

Prove that $x+2x^2+3x^3+4x^4+...+nx^n = \frac{nx^{n+2}-(n+1)x^{n+1}+x}{(x-1)^2}$ I see that this can be written as $$\sum_{n=1}^n nx^n = n\sum_{n=1}^n x^n$$ $$\sum_{n=1}^n x^n = ...
0
votes
2answers
37 views

A simple algebra question

What is the simplest method for solving $x(x+a)=a$ or $x=a/(x+a)$ for a. I think there's a trick for solving algebra questions like this. This problem comes up when deriving the sum of a geometric ...
2
votes
3answers
69 views

Solving Recurring Relations

Can you please help, my son has been trying for over two hours now to solve the following: A sequence of terms $\left\{u_n\right\}$ is defined for $n\geq 1$, by the recurrence relation: ...
0
votes
0answers
30 views

Quick simplification strategy for $\binom{3}{2}p^2(1-p) \le \sum_{k=3}^{5}\binom{5}{k}p^k(1-p)^{n-k}$

What is a quick simplification strategy to solve the following expression for $p$ by hand? (or less preferably, by a TI83/86 calculator). $$\binom{3}{2}p^2(1-p) \le ...
0
votes
1answer
33 views

The equation $f(x)= \frac{3^x+1}{2}$ for all positive integers x

The equation $f(x)= \frac{3^x+1}{2}$ for all positive integers $x$ generates a sequence such that the difference of every consecutive f(x) forms another say g(x) such that $g(x) = 3^{x-1}$. What I did ...