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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

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. This can be useful when evaluating (in)finite series or determining a closed form for a recurrence relation.

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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, 931, 469, 964, 248, 842 but then $3+(n-1)d$…
Riddhiman
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Special case application of geometric progression not adding up

Consider $N_1 = \sum_{k=1}^K kr^k$ and $N_2 = \sum_{k=0}^K kr^k$ (assuming $r \neq 1$). These two equations are equivalent since the first term in $N_2$ will always be zero due to the multiplication by $k$. However, solving now: $N_1 = \sum_{k=1}^K…
Allan_ZA
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Can't figure out why $\sum_{k=1}^{\infty }\frac{1}{3}\times 2^{| -2k+1 |} = 2/9$

1) In my guide I have that: $$ \sum_{k=1}^{\infty }p_{k}=\sum_{k=1}^{\infty }\frac{1}{3}\times 2^{-2k+1} = 2/9$$ I can't figure out what I have this. 2) Plus, the original probability function was $$ p_{k}=\frac{1}{3}\times 2^{-\begin{vmatrix}…
Limanido
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Why the geometric series of a random variable that takes all $\mathbb Z$ numbers include $p_{0}$+constant?

As I don't have at least 10 reputation so I cannot post formulates made by another web page like codecogs. My question is that I need to calculate the geometric progression of (you can post this codeon codecogs and see what Im trying to ask) $$ …
Limanido
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What is the largest number of members that a geometric progression can have in this problem?

What is the largest number of members that a geometric progression can have, whose members are various natural numbers, greater than $210$ and less than $350$? I'm not really sure what method to use solving this. sorry i know it doesn't look that…
young
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Find term basing on point.

I'm making a level calculation system, based on geometric progression, with initial term $a_1=100$ and ratio $r=2$. So, we have an equation for min. $x$ and max. $a$ XP values per level $n$: $a_n = a_1*r^{n-1} \\ x_n = a_{n-1} = a_1*r^{n-2}$ I need…
nesclass
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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 receives $y$ - number of players $n$ -…
Yuri Vaskovski
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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 $a\frac{1-q^{n+1}}{1-q}$ I got $\frac{(1+x^2)^{n+1} -…
E. Shcherbo
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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 internet.
Dante
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Geometric Progression Question with terms to infinity

I'm having a problem with the question stated below. I stumbled upon it during my revision And I was hoping one of you guys could help me solve it and better yet Understand how to go about it. **A geometric Progression has the first term a, common…
RodrigoArts
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Exercise about geometric progession

I have an statement that says: Find 3 numbers in P.G, if this numbers add up $12$ and its product is $-216$ I tried this: The $r = \frac{\pm\sqrt{5} - 3}{2}$ And when i replace the $r$, it not give $12$ in the addition
ESCM
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Geometric progression exercise

I have this statement: Three numbers add up $155$ and its product is $15625$, ¿What are the terms? I tried this: Ok, three numbers add up $155$ is: $a_k + a_k * r + a_k * r^2 = 155$ And, its product is 15625: $a_k * a_k * r * a_k*r^2 = 15625$, i…
ESCM
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Prove it is a geometric progression.

Question Prove that if for an exponential function $ y = a^x (a>0;a \neq 1)$ the value of the argument $x=x_n(n=1,2,...)$ form an arithmetic progression, then the corresponding values of the function $y_n=a^{x_n}(n=1,2,...)$ form a geometric…
Pratik Phadatare
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Prove a sequence is geometric

Suppose I am given that the sum of the first $2n$ ($n$ is a positive integer) terms of a sequence $u_{1},u_{2},...$ is given by $\frac{3}{10}-\frac{1}{10(3)^{2n-1}}$ and I need to show that the sequence is geometric. My question : Is it possible to…
LanaDR
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Dilemma in a problem on GP

PROBLEM If a, b, c, d are four distinct positive quantities in G.P., then show that a + d > b + c Solution in the book: A.M. > G.M. for the first three terms $(a+c)>2b$; since $ac=b^2$ ................(A) Similarly, for the last three…
Marble
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