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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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Can you prove that $\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}.... = \frac{1}{n-1}$?

I'm a student and, while playing with my calculator, find out that: $$\frac{1}{n}+\frac{1}{n^2}+\frac{1}{n^3}.... = \frac{1}{n-1}$$ is it a well defined equivalence and what is its name, is there a proof for that? if we put it this…
user177880
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Solve the following equation for x :

$$(x + 1)^{63} + (x + 1)^{62}(x−1) + (x + 1)^{61}(x−1)^{2} + . . .+(x−1)^{63}= 0$$ My approach was: Its a GP with $$r= \frac{(x-1)}{(x+1)}$$ Then with the expression: $ ar$$n-1$$=(x-1)$$63$ Plugging value of r and a it results to…
Amar30657
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Coefficient to create a series whose sum is a given one

I realize that my title is not clear but I currently don't master math terms well enough to write a better title or to enter meaningful tags. Any correction to my question, tags and title would be very welcome and I hope to learn fast. This is my…
SantiBailors
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Eligibility for geometric progressions containing few terms

Question: Does there exists a GP containing 8, 12 and 27 as three of its terms? If it exists how many such progressions are possible? What I have tried: First of all I have made three equations as follows Here 27, 12, 8 are $p^{th},q^{th},r^{th} $…
Kartik Watwani
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Minimum distance between $2^n$ and $3^m$

I’m looking for the minimum distance between any two members of the geometric progressions 2, 4, 8,… and 3,9,27,… It seems like the pair of numbers which has the minimum distance is (2,3). Can you help me find a proof? Also what if one of the…
user499369
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Find the sum of $5.5+55.55+555.555..$ up till n terms?

Find the sum of $5.5+55.55+555.555..$ up till n terms? My attempt: $ 5.5+55.55+555.555 ... $ $ 5(1.1+11.11+111.111...) $ $ \dfrac{5}{9} \times 9(1.1+11.11+111.111..) $ $ \dfrac{5}{9} (9.9+99.99+999.999...) $ $ \dfrac{5}{9}…
user876009
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geometric figure illustrating the geometric progression 1,r,r2,r3,r4,r5,...

Here is a geometric figure illustrating the geometric progression 1,r,r2,r3,r4,r5,…: from here What is happening here? With a being the unknown side of the 2nd (=1-r-triangle) and b the unknown side of the third triangle, I know 1/b=a/r=r/x. How do…
youdontknowme
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How to formulate a GP according to this question?

Q. Each term of a geometric progression is $\frac{1}{x} th$ of the sum of all the terms following it. Find the commmon ratio of the progression in terms of $x$. A. $\frac{x-1}{x}$ B. $\frac{x}{x+1}$ C. $\frac{x+1}{x+2}$ D. None The problem is how to…
shreyassps
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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 progression formula. I know my $a_1$ is 2400. I…
IncognitoBatman
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Application of geometric progression

I have an exercise, which says: Determine 4 numbers in geometric progression, such that the first 2 numbers add 9, and the last two numbers add up to 36. I know all the formulas and I tried everything, but I can not get the result. I thought that…
ESCM
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Can question have multiple interpretations?

In Simple Terms Annie and Ben are both doing the question... A 100m cliff erodes by 2/7 of its height each year. (a) What will the height of the cliff be after 10 years? Annie believes the answer is 100*(5/7)^9 because using the geometric…
Bradman175
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Fermat's Theorem involving GP

Question:It is known by Fermat's Theorem that $n^p-n= M(p)$= a multiple of $p$,if $p$ is a prime number and $n$ is a prime to p.If $n$ is a prime number which divides neither $a,b$ nor $a+b$ ,prove…
Kartik Watwani
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Find the sum of the expression below

Calculate: $\dfrac{1}{2}+\dfrac{2}{2^²}+\dfrac{3}{2^3}+\dfrac{4}{2^4}+...=?$ I…
peta arantes
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Determine the sum value of the first terms of that sequence.

Consider a geometric progression in which the common ratio is a non-zero natural number, knowing that the logarithm of the nth term at the base equal to the common ratio of the sequence is equal to 6, that the logarithm of the product of the…
peta arantes
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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 example 240.353 What's the formula to determine how…
Codem
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