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Questions tagged [geometric-progressions]

A geometric progression is a sequence of numbers such that the quotient of any two successive members of the sequence is a constant called the common ratio of the sequence

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How to get sum of $\frac{1}{1+x^2}+\frac{1}{(1+x^2)^2}+…+\frac{1}{(1+x^2)^n}$ using mathematical induction

Prehistory: I'm reading book. Because of exercises, reading process is going very slowly. Anyway, I want honestly complete all exercises. Theme in the book is mathematical induction. There were ...
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2answers
24 views

Relation between two series

Consider the two series , A=Σ(2ⁿ/n!) from 1 to ∞. and, B=Σ(4ⁿ/n!) from 1 to ∞. What is the relationship between them?( If any) I think the exponential series might come in handy but the numerator ...
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In a geometric progression, we know the partial sums $S_2 = 7$ and $S_6 = 91$. Find $S_4$.

In a geometric progression, $S_2 = 7$ and $S_6 = 91$. Evaluate $S_4$. Alternatives: 28, 32, 35, 49, 84. Here's what I tried so far: $$ S_2 = \frac{a_1(1-r^2)}{1-r} \implies 1-r = \frac{a_1(1-r^2)}{7}...
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Finding all continuous function which maps any sequence in geometric progression to another geometric progression

Find all continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for any geometric progression $x_n$ the sequence $f(x_n)$ is also a geometric progression. I tried first by taking constant ...
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26 views

Summation of series to n terms in trigonometry of complex numbers

The question says that: Sum the series I have solved the answer as follows: As the above picture, I don’t know what should I do after the step. The question asks to solve the problem using ...
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1answer
26 views

What kind of sequence/progression is this? What will be the answer to the question?

There are 20 urns such that the first urn contains 5 balls, the second contains 10 balls and in general the $k^{th}$ urn contains $2k + 1$ balls more that that in $(k - 1)^{th}$ urn. Then what is the ...
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5 numbers in AP, GP and HP.

The Question: Consider 5 numbers $a_1,a_2,a_3,a_4,a_5$ such that $a_1,a_2,a_3$ are in AP, $a_2,a_3,a_4$ are in GP and $a_1,a_4,a_5$ are in HP. Then find whether $\ln a_1,\ln a_3, \ln a_5$ are in AP,GP ...
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39 views

Upper bound on partial sum of a geometric series

For a geometric series $1,r,r^2,\dots$ with integer $r \ge 2$, denote the $k^{th}$ partial sum by $\Sigma_k = \sum_{i=1}^k r^{i-1} = (r^k - 1)/(r-1)$. Is it true that $\Sigma_k < 2^{\lfloor \log_2 ...
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The ordinary iteration method converges faster than any geometric progression.

I have gotten stuck trying to prove that iteration method converges faster than any geometric progression. Background: Assume that the function $g$ is continuously differentiable. Let $x^*$ be the ...
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To find common difference of a logarithmic AP.

Ques: If $a,b,c$ are in GP and $\log_ba,\log_cb,\log_ac$ are in AP. Then find the common difference of AP. Here's what I did: $\Rightarrow b^2=ac$ $\Rightarrow 2\log b=\log a+\log c$ i.e. $\log a,\...
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Find the original three terms

$x$, $y$, and $\frac3{2x}$ are non-zero terms in an arithmetic progression. If the third term is increased by $1$, the three terms now form a geometric progression. Find the original three terms. ...
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1answer
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Geometric Progression via 10% consecutive term

Determine the sum of the geometric progression with each consecutive term being 10% larger than the previous term and the first term is 2400. I tried solving it like this below via the Geometric ...
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1answer
39 views

How do I prove this relationship between positive terms of a G.P.?

$a$, $b$, $c$, and $d$ are positive terms of a G.P. This is the relationship I'm trying to prove: $$\frac1{ab} + \frac1{cd} > 2 \left(\frac1{bd} + \frac1{ac} - \frac1{ad}\right)$$ This ...
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Is $1111111111111111111111111111111111111111111111111111111$ ($55$ $1$'s) a composite number?

This is an exercise from a sequence and series book that I am solving. I tried manipulating the number to make it easier to work with: $$111...1 = \frac{1}9(999...) = \frac{1}9(10^{55} - 1)$$ as ...
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1answer
37 views

How to solve for x when function can't be inverted

The sum of the first three members of a geometric progression is equal to 42. Those same members are, correspondingly, the first, the second and the sixth member of an arithmetic progression. The task ...
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How to calculate the value of a coin? [closed]

I do not need a specific answer, but a global formula that solves this equation. There is a coin that costs 100\$ (or 5\$ or \$25 etc.) Each year, it steadily increases by 10% (or 1% or 4% etc.) How ...
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How to solve $S = x + xn + xn^2 + \cdots + xn^{y-1}$ for $n$

I need to come up with a formula to calculate the coefficient from this formula $$S = x + xn + xn^2 + \cdots + xn^{y-1} \tag{1}$$ Variables: $S$ - total prize pool $x$ - amount the last place ...
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find the first term of the series?

The sum of an infinite geometric series of real numbers is $14,$ and the sum of the cubes of the terms of this series is $392$. What is the first term of the series? My attempt: Let the series be $\...
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Prove a geometric sequence a, b, c from the arithmetic progression $1/(b-a)$, $1/2b$, $1/(b-c)$

The given task is: The following forms an arithmetic sequence: $$\frac{1}{b-a}, \frac{1}{2b}, \frac{1}{b-c}.$$ Show, that $a, b, c$ forms an geometric sequence. It's easily enough to understand that $...
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Find the range of values of $a$ if $p(x)=x^2-ax+b$, and $p(0)$, $p(1)$, $p(2)$ form a non-zero geometric progression

Find the range of values of $a$ if $p(x)=x^2-ax+b=0$ and $p(0)$, $p(1)$, $p(2)$ form a non-zero geometric progression.
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If $(m+1)$ th ,$(n+1)$ th,$(r+1)$ th terms of a non constant AP are in GP and $m,n,r$ are in HP,

If $(m+1)$ th ,$(n+1)$ th,$(r+1)$ th terms of a non constant AP are in GP and $m,n,r$ are in HP,then prove that the ratio of first term of the AP to its common difference is $\frac{-n}{2}$ Let$(m+1)$ ...
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Sum of n terms of a G.P. confusion in formulas

If we have to find the sum of n terms of a G.P. then we have two formulas for it (1) $a(1-r^n)/(1-r)$ and (2) $a(r^n-1)/(r-1)$. Now I know how the (1) has been derived but dont know about the (2)(is ...
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Sum of Geometric progression?

A man deposits $200 at the beginning of every year into a bank account at a compund interest rate of 3%per annum. Find out how much he has at the end of 10th year to the nearest dollar? I honestly ...
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1answer
82 views

How to prove geometric progression

The given series is: $$6+3*2^{1-n}$$ Prove that this series is geometric progression. What is $a_n$ in this series? And also show if the series is convergent. I tried to: So we know that the sum of ...
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4answers
51 views

Why $b^2=a\cdot c$ is being used when the terms are not consecutive?

Okay most of us know that if $a,b,c$ are three terms of a sequence then they are in G.P. if $b^2=a\cdot c$. But isnt this formula $b^2=a\cdot c$ is only applicable when the terms are consecutive. ...
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1answer
39 views

How Can I Represent These Progressions in Sigma Notation?

I would like to represent the following finite progressions in sigma notation: $Finding\ the \ n^{th} \ term \ of \ a \ geometric \ progression$: $a_n=a_1(r^{n-1})$, where $a_1$ is the first time and ...
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A proof on multinomial roots

If $x_1,x_2,...,x_{n-1},x_n$ be the roots of the equation $$1 + x + x^2 + ... + x^n = 0$$ and $y_1,y_2,...,y_{n},y_{n+1}$ be those of equation $$1 + x + x^2 + ... + x^{n+1} = 0$$ show that $$(1-x_1)(1-...
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1answer
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How to relate two series(GP and AP) using a positive real number, who have nothing in common?

Here is a question from my book: Given a GP and an AP with positive terms $a,a_1,a_2,a_3...a_n$ and $b,b_1,b_2,b_3,...b_n$ respectively. The common ratio of the GP is different from $1$. Then ...
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Double summation with improper integral

So my friend sent me this really interesting problem. It goes: Evaluate the following expression: $$ \sum_{a=2}^\infty \sum_{b=1}^\infty \int_{0}^\infty \frac{x^{b}}{e^{ax} \ b!} \ dx .$$ Here is ...
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Annuities with payments in geometric progression

I am having trouble understanding how to solve problems with varying annuities. There is this problem I was given as a homework which I can't figure out. Barry presently has 2.9 million dollars in ...
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What is the closed form of $\sum_{j=i}^{n} {j}$?

How can I get a closed form from a summation like this? $$ \sum_{j=i}^{n} {j} $$ I don’t know how to proceed since the base of the summation is a variable.
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Solving summations by upper bound - lower bound + 1

I’ve seen many different summations that are solved by the upper bound minus the lower bound plus one. For example: $$ \sum_{i=1}^{n} {1} = n-1+1 $$ May I always resolve summations in this way or ...
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3answers
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How to compute the time complexity of a triple nested loop represented by $\sum_{i=1}^{2n} \sum_{j=1}^{n} \sum_{k=j}^{n} i-j$

var r = 0; for(var i=1; i<=2*n; i++) { for(var j=1; j<=n; j++) { for(var k=j; k<=n; k++) { r = r + (i - j); } } } I'm trying to use ...
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Find minimum of $ [ x^{(\ln y-\ln z)} + y^{(\ln z-\ln x )} + z^{(\ln x-\ln y)} ]$

If $$x\gt0,y\gt 0, z\gt 0$$ then find the minimum value of $$ \left[ x^{(\ln y-\ln z)} + y^{(\ln z-\ln x )} + z^{(\ln x-\ln y)} \right]$$ Today I came across this question ( maybe it's from ...
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1answer
32 views

Resolving dependencies of nested dependent summations

If I have two or more nested summations in which the inner ones depend on the outer ones, how could I “remove” or resolve the dependencies? In this case, for example: $$ \sum_{i=1}^{n-1} \sum_{j=i+1}...
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4answers
76 views

There are $4$ numbers. First $3$ make an arithmetic progression. Last $3$ make a geometric progression.

There are four numbers. The first three make an arithmetic progression, and the last three make a geometric progression. The sum of first and last number is $37$. The sum of middle numbers is $36$. ...
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prove that $q_n = p_n \, \sin(\frac{t}{2^{n-1}})$ is a geometric progression

Given $$p_n = \prod_{n}^{p=1} \cos\left(\frac{t}{2^{n-1}}\right) \hspace{3mm} \text{and} \hspace{3mm} q_n = p_n \, \sin\left(\frac{t}{2^{n-1}}\right).$$ How to prove that $(q_n)_{n}$ is a geometric ...
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A nos of squares are described whose perimeter are in GP.Then their side will be in : a)AP b)GP c)HP d)Nothing can be said e)none of above

the process is as below:- let perimeter be, 4a ,8a,16a,32a... sides will be a,2a,4a,8a.. and this shows that sides are in gp so i want to ask here is that if i consider the side of the square as a ...
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118 views

Suppose the sides of a triangle form a geometric progression with common ratio r. Then what interval does r lie in? ( Options are down below) [closed]

Options are A ( 0, (-1+✓5)/2 ] B ( (1+✓5)/2 , (2+✓5)2 ] C ( (-1+✓5)/2 , (1+✓5)/2 ] D ( (2+✓5)/2 , Infinity ) Im just a 12 th grade . So i request answer with explanation please.
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Find the common ratio, if the sum of the first $8$ terms in a geometric progression is equal to $17$ times the sum of its first $4$ terms [closed]

Find the common ratio, if the sum of the first $8$ terms in a geometric progression (GP) is equal to $17$ times the sum of its first $4$ terms. So far I have got $$a+ar+ar^2+ar^3+ar^4+ar^5+ar^6+ar^7=...
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Sum of Geometric Series Formula [duplicate]

I just need the formula for the sum of geometric series when each element in the series has the value $1/2^{j+1}$, where $j = 0, 1, 2, \ldots, n$. Please help. Someone told me it is: $$S = 2 - \frac{...
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1answer
79 views

Evaluating the series $\sum_{k=1}^\infty \frac{2\times 3^k}{4^{2k+1}}$

$$\sum_{k=1}^\infty \frac{2\times 3^k}{4^{2k+1}}$$ Hi all, I finally am getting the hang of MathJax (sort of) thank goodness! I was hoping for some help on a problem involving series. I am stuck ...
2
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3answers
53 views

Let $R_n = \underbrace{11\ldots1}_{n}$. Prove that if $(n, m) = 1$, then $(R_n, R_m) = 1$.

Let $R_n = \underbrace{11\ldots1}_{n}$. 1) Prove that if $n \mid m$, then $R_n \mid R_m$. 2) Prove that if $(n, m) = 1$, then $(R_n, R_m) = 1$. I have solved only the first task: $$m = nk \...
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60 views

Calculate the sum of geometrical progression

I have the following progression $$ \frac{1}{1+x^2} + \frac{1}{(1+x^2)^2} + ... + \frac{1}{(1+x^2)^n} $$ I have that $a=\frac{1}{1+x^2}$ and $q=\frac{1}{1+x^2}$, then using $a\frac{1-q^{n+1}}{1-q}$...
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0answers
46 views

Sequences and geometric progressions

Any sequence of natural numbers that contains an infinite arithmetic progression (AP) must have a positive lower density and this alone rules out many candidates (squares, primes, etc.). On the other ...
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2answers
52 views

What is the general formula for expansion of 1 + x^(odd number) in terms of (1+x)

I believe there's formula to write $1+x^{2n+1}$ in terms of $(1+x)(\cdots)$ just like how $1+x^3$ can be written as $(1+x)(x^2 -x +1)$. I am not able to find explanation of it anywhere over the ...
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1answer
113 views

How to explain mortgage monthly payment formula using school math?

$P = L*\frac{x*(1+x)^n}{(1+x)^n - 1}$ where P - monthly payment L - loan amount x - monthly interest rate n - number of payments Here is in ...
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2answers
103 views

$a, b, c$ form a geometric sequence and $\log_c a, \log_ b c, \log_a b$ form an arithmetic sequence.

The common difference of the arithmetic sequence can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ So far, I rearranged the sequences to be 1: $a$,...
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2answers
75 views

What is $\sum_{i = 0}^{n} (2^{ki})$? [closed]

What is the result of $\sum\limits_{i = 0}^{n}(2^{ki})$ Is it $\frac{1-2^{kn}}{1-2^k}$?
3
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1answer
138 views

rank of a rectangular Vandermonde matrix

Let the $m\times (n+1)$ rectangular Vandermonde matrix be $V$. More specifically, the matrix $V$ has the following form. $V=\begin{pmatrix} 1 & a_1 & \cdots & a_1^{n} \\ 1 & a_2 &...