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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Partial sum of geometric series of a Ceiling Function $\sum_{i=1}^n\lceil k\cdot(1.5)^i\rceil$ [closed]

I need to find a closed form of $n$-partial sum: $$\sum_{i=1}^n\lceil k\cdot(1.5)^i\rceil$$ Where $k \approx 1.08151366859$ and is irrational. Alternatively, sum of first $n$ elements of sequence ...
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69 views

How to sum $\sum_{x=0}^{N-1} \cos{\left( \frac{2\pi x}{N} \right)}^{2}$?

I've tried solving it, but I'm not familiar with these kind of progressions and I couldn't get to a result. Also, I've found that the result is just $N/2$, but I don't get why would that be the answer,...
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1answer
81 views

How do i verify that $272727…2727$ ($100$ digits) can or cannot be written as a perfect square?? [duplicate]

I've been stuck on this question. I tried writing the number as as geometric progression plus $$2((10^{100}-1)/9)+5+5.10^2+5.10^4...$$ Got stuck in there.
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2answers
81 views

How many terms of the geometric sequence $2, 8, 32, 128,\dots $are required to give a sum of $174,762$?

How many terms of the geometric sequence $2, 8, 32, 128,\dots $are required to give a sum of $174,762$? My attempt $a = 2$ (the first term) $r = 4$ (the common ratio) $S_n = 174,762$ (sum to $n$ ...
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1answer
29 views

Calculate Geometric Progression with logarithm

I'm trying to find the following summation but cant seem to find the answer. Would really appreciate any help from here! $$\sum_{i=0}^{k}(4/3)^{i}\log(n/3^i)$$
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1answer
106 views

Find $a_1+\frac{a_2}{2}+\frac{a_3}{2^2}+\cdots\infty$

A sequence $\left\{a_n\right\}$ is defined as $a_n=a_{n-1}+2a_{n-2}-a_{n-3}$ and $a_1=a_2=\frac{a_3}{3}=1$ Find the value of $$a_1+\frac{a_2}{2}+\frac{a_3}{2^2}+\cdots\infty$$ I actually tried this ...
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What's geometric about a “geometric progression”? [duplicate]

An arithmetic progression is $a+0b, a+1b, a+2b, ..., a+nb$ A geometric progression is $ab^0, ab^1, ab^2, ..., ab^n$. Multiplication is arithmetic, so why is a geometric progression not also an "...
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72 views

The common difference is equal to the common ratio.

Four numbers are in A.P. The first, the second and the fourth are in G.P. Find the numbers if the common difference is equal to the common ratio. Let the terms of the A.P. be $a_1,a_1+d,a_1+2d,a_1+3d$ ...
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35 views

Can the values of the expressions $\frac{1}{\sqrt{2a+1}},\frac{1}{\sqrt{2a-1}},3(a^2-19)\sqrt{2a-1},18\sqrt{2a+1}$ be in G.P.?

Can the values of the expressions $\dfrac{1}{\sqrt{2a+1}},\dfrac{1}{\sqrt{2a-1}},3(a^2-19)\sqrt{2a-1},18\sqrt{2a+1}$ be in geometric progression (in the given order)? I am confused by the fact that ...
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1answer
19 views

Given sum of infinite G.P, find the common ratio.

Sum of an infinite $G.P$ is $2020$. Each term of this $G.P$ is squared to make a new series whose sum is $20200$. If the common ratio of the original $G.P$ is ${a\over b}$ where $gcd(a,b)=1$, evaluate ...
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60 views

The $2$nd, the $1$st and the $3$rd term of an arithmetic progression form a geometric progression

An arithmetic progression is given with a common difference $\ne0.$ The $2nd$, the $1st$ and the $3rd$ term of the ap form a geometric progression. Find the common ratio. So we have the ap: $a_1,a_1+d,...
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Can I write e(mathematical constant) in fraction? [closed]

Proof of e can be in fraction e is 2.718281828..... so as you can see, there is a pattern. So, I break it down into 2.7 + 0.01828 + 0.0000001828 ..... and then I use sum of infinity to get the ...
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1answer
81 views

Problem with a Arithmetico-Geometric Series

Good afternoon to everyone, I have the following question: What does the arithmetico-geometric series: $$S = \sum^{\infty}_{n=1} ne^{-nrt}$$ Converge to? ($r > 1$, $t > 1$) I tried to break it ...
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32 views

A progression question with elements of geometry…

$ABCD$ is a square with area $1 cm^2$. Draw an isosceles right $\triangle CDE$, right angled at $E$. Draw another square $\square DEFG$, then draw an isosceles right $\triangle FGH$, right angled at $...
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2answers
40 views

Polynomial with geometric progression [closed]

If the roots of an equation with real coefficients $ax^3+bx^2+cx+d=0, a\ne0$ form a geometric progression, prove $ac^{3}=db^{3}$ I have no idea how to get started, so if it can a few hints?
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1answer
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GP and combinations divisibility

Is the number : ${100 \choose 0}\cdot2^0 + {100 \choose 1}\cdot2^1+{100 \choose 2}\cdot2^2+{100 \choose 3}\cdot2^3 \ldots + {100 \choose 100}\cdot2^{100}$ divisible by 3? I tried looking at the first ...
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1answer
15 views

Determining the position of a random non-natural positive rational number in a geometric progression

I multiply 100 by 1.05. I get 105. I multiply 105 by 1.05. I get 110.25. I multiply 110.25 by 1.05. I get 115,7625 and so on. If I choose a fully random non-natural positive rational number, for ...
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1answer
24 views

Remainder for a GP

What is the remainder of : $\sum_{i=0}^{2019} 3^i$ divided by $3^4$ ? I know that $$\sum_{i=0}^{2019} 3^i = \frac{3^{2020}}{3-1} = \frac{3^{2020}}{2} $$ so $$\frac{\sum_{i=0}^{2019} 3^i}{3^4} = \...
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2answers
65 views

$(ab+bc+ca)^3=abc(a+b+c)^3$, prove that $a,b,c$ are in $G.P.$ [duplicate]

Suppose $a,b,c$ are non-zero real numbers such that $$(ab+bc+ca)^3=abc.(a+b+c)^3$$ Prove that $a,b,c$ must be terms of a $G.P.$ I simplified this equation too $$(ab)^3+(bc)^3+(ca)^3=abc.(a^3+b^3+c^3)$...
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67 views

One root common to $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$

If three distinct numbers $a,b,c$ are in GP, and the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root, then which of the following statements is correct? $1.$ $d,e,f$ are in GP. $2.$ $...
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3answers
71 views

What is the relationship between A and B?

We have $0<b≤ a$, and: $$\underbrace{\dfrac{1+⋯+a^7+a^8}{1+⋯+a^8+a^9}}_{A} \quad \text{and} \quad \underbrace{\dfrac{1+⋯+b^7+b^8}{1+⋯+b^8+b^9}}_{B}$$ Source: Lumbreras Editors It was my strategy: ...
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2answers
142 views

Why is the sum to infinity of a geometric distribution equal to 1?

I know that the geometric distribution follows the rules of a geometric progression thus by using the sum to infinity formula (which I know its proof and is really convinced by it), $$\frac {a}{1-r}$$ ...
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1answer
40 views

Summing the first $n$-terms of the series whose general term is $nx^{n-1}$

I suppose several of you know some fancy ways to establish the formula for the sum of the first $n$ terms of the geometric series $$1+x+x^{2}+x^{3}+ \ldots $$ Can you share below some of your fave ...
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2answers
39 views

Finding all possible values of $d$ given that the sum of the reciprocals of a polynomial is $17$

Let $p(x) = x^5 - 833x^4 + ax^3 + bx^2 + cx + d$ such that the roots of $p(x)$ are in geometric progression. If the sum of the reciprocal of the roots is $17,$ determine all possible values of $d.$ I ...
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1answer
41 views

Prove the inequality without using the concept of Arithmetic and Geometric mean inequality

$\displaystyle ( 2m+1) r^{m}( 1-r) < 1-r^{2m+1}$ where $r<1$ and m is positive integer. I can prove it by concept of arithmetic and geometric mean inequality. But I am curious to know whether ...
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3answers
46 views

How do we calculate the following limit?

The limit to be found is:- $$ \lim_{x \to \infty} \frac{2+4+...+2^x}{2^x +1}$$ My attempt:- Taking $ 2^x $ common from numerator and Denominator, $$ \lim_{x \to \infty} \frac{(2/2^x)+(4/2^x)+...+1}{1 +...
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1answer
581 views

Find the first term and the common ratio of an infinite geometric series

Find the first term and the common ratio of an infinite geometric series whose sum is $5$ and such that each term is $4$ times the sum of all the terms that follow it. I used $a_{1}r^{3}=\frac{4[a_{1}...
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2answers
35 views

Congruent sets of an arithmetic sequence and a geometric sequence

Suppose we have a $a,d,$ and $q$ such that $a \neq 0, d \neq 0.$ Then, let $M = \{a, a + d, a + 2d\}$ and $N = \{a, aq, aq^2\}.$ Given that $M = N,$ find the value of $q.$ (A) $\frac12$ (B) $\frac13$ (...
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1answer
260 views

Total area of infinite circles nested in an equilateral triangle.

Given that the radius of the bigger circle is 1, what is the total area of the infinite circles in the picture above? I know how to solve part of the problem, following the steps of this site. But the ...
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1answer
249 views

Finding the common ratio of the geometric progression

There is a question in my book stating that A geometric progression consists of an even number of terms. If the sum of all the terms is five times the sum of terms occupying odd places then find the ...
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4answers
62 views

Identifying relation between numbers based on equation relating them

If $a^2 + b^2+16c^2=2(3ab+6bc + 4ac)$ , where $a,b,c$ are non zero numbers. Then $a,b,c$ are in __________? 1. Harmonic progression 2. Geometric progression 3. Arithmetic progression 4. None of these ...
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What's the name for sequences sitting between geometric and arithmetic, i.e. whose recurrence relation is of the form $ax+b$?

What's the name for sequences sitting between geometric and arithmetic? E.g. let $x_{n+1}=ax_n+b$ I can't find a general name for these. These sequences may also be a Lucas Sequence but that's a ...
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1answer
32 views

If $n \in \mathbb N$, find $\sum(-1)^r \binom{n}{r}\left(\frac {1} {2^r}+\frac {3^r} {2^{2r}}+\frac {7^r} {2^{3r}} + \cdots \text{m terms}\right)$ [duplicate]

If $n \in \mathbb N$, find$$\sum(-1)^r \binom{n}{r}\left(\frac {1} {2^r}+\frac {3^r} {2^{2r}}+\frac {7^r} {2^{3r}} + \frac {15^r}{2^{4r}} + \cdots \text{upto m terms}\right)$$
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3answers
97 views

Geometric sequence problem including sum of the numbers

Numbers: $a,b,c,d$ generate geometric sequence and $a+b+c+d=-40. $ Find these numbers if $a^2+b^2+c^2+d^2=3280$ I tried this problem and I have system of equations which I can't solve. I think there ...
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0answers
35 views

Weighted sum with geometric decreasing weights

First time here, but I'm in a sorta challenge. So, let's say we have a sequence $x_i$ with $i=1,2,...,n$ such that $x_i\geq x_j \forall 1 \leq i \leq j \leq n$. Let's define a value $S$ for the ...
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2answers
126 views

Showing that $a$, $b$, $c$, $d$ are in geometric progression iff $(a^2+b^2+c^2)(b^2+c^2+d^2)=(ab+bc+cd)^2$

If the real numbers $a$, $b$, $c$, $d$ are in geometric progression, show that $$ \left(a^{2}+b^{2}+c^{2}\right)\left(b^{2}+c^{2}+d^{2}\right)=(a b+b c+c d)^{2} $$ Prove that the converse also holds. ...
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1answer
76 views

Question on probability, please help me understand the intuition behind it

I was going through a set of problems at codeforces.com, I came across a very interesting question (link: https://codeforces.com/contest/312/problem/B): Question: SmallR is an archer. SmallR is ...
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1answer
26 views

Is there a closed formula for the sum of a geometric progression with binomial coefficients? [closed]

The title asks it all. $$\sum_{i=0}^n{n\choose i}x^{i+1}=?$$
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2answers
54 views

Find the common ratio of the progression. [closed]

A geometric progression has 625 as the first term. The product of its first 3 terms is equal to the product of its first 6 terms. Find the common ratio of the progression.
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1answer
66 views

Trigonometry of geometric and arithmetic progressions

In $\triangle ABC$, $\ AB=AC$, $\ BD=DC$, and $\ BE>CE\ $. If $\ \tan\angle EAC$, $\ \tan\angle EAD$, $\ \tan\angle EAB$ are in geometric progression, and $\ \cot\angle DAE$, $\ \cot\angle CAE$, $\ ...
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1answer
50 views

Finding K value?

Suppose that a geometric progression (GP) $1, q, q^{2}, \ldots$ (where $|q|<1)$ is to be constructed such that every term of this GP is a constant multiple (say $k$ ) of sum of all the subsequent ...
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40 views

Geometric Sum Formula

In the textbook I'm working from, the formula for the sum of a geometric progression is given as $S^n=\frac{a(1-r^n)}{1-r}$ It then adds that the formula may also be written as $S^n=\frac{a(r^n-1)}{...
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2answers
82 views

Geometric Series to approximate power series expansion

I have been stuck working through the following derivation for a while now (for context the problem is a Central Limit Theorem proof in Stochastic Processes: Theory For Applications, specifically the ...
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1answer
54 views

Geometric series in proof of Stirling's Formula

I am working through a proof of Stirling's Formula in Feller's An Introduction to Probability Theory and it's Applications and am stuck at equation 9.10, where he make a comparison with a geometric ...
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1answer
28 views

All AP of natural numbers starting $3$ which has a $3$ digit sum whose digits are in non constant GP t

The question is: Find all Arithmetic progression of natural numbers starting with $3$ whose sum is a $3$ digit number whose digits are in non constant GP. I tried that the sum could be $124, 421, 139,...
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2answers
59 views

How can $(1+\rho^2+\rho^4+\rho^6+\rho^8+…)= \frac{1} {1-\rho^2}$? [closed]

$$(1+\rho^2+\rho^4+\rho^6+\rho^8+...)= \frac{1} {1-\rho^2},$$ where $|\rho| <1$ I don't see how this simplifies. Thanks.
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0answers
30 views

Infinite Harmonic Progression

Today, I was handing a lot with Progressions. It came to my mind that there are infinite AP and GP. for example :- An Arithmetic Progression with First term 2 and common difference 5 looks like the ...
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2answers
29 views

How to derive the closed form of the series [closed]

The closed form is $2^r$ and the series is $1+2+4+8+\ldots$.
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1answer
41 views

Solving sum of Arithmetic Geometric Progression in different ways and getting different answers

I have a progression that goes like this S = $\frac{3}{19}+\frac{33}{19^2}+\frac{333}{19^3}+\frac{3333}{19^4}... \infty$ I tried solving this sum in this way Multiply both sides by $3$ $3S$ = $\...

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