# 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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### geometric progression point distribution when one extreme point is negetive

How to generate 20 points from -0.01 to 100 which are geometrically equal in separation means if I want to plot in log scale $\log d_2 - \log d_1 = \log d_3 - \log d_2$ where $d_1$, $d_2$, $d_3$ ...
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### Generalized Arithmetico-Geometic series with a possible application

While playing with Arithmetico-Geometric progression formula(i.e $$\sum_{k=1}^{n}(a+(k-1)d)y^{k-1} = \frac{a-[a+(n-1)d]y^n}{1-y} +\frac{1-y^{n-1}}{(1-y)^2}yd$$ I realized it could be generalized as: ...
1 vote
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### Sum of the series of numbers consisting of AP and GP both.

Find the sum of all the terms, if the first $3$ terms among $4$ positive $2$ digit integers are in AP and the last $3$ terms are in GP. Moreover the difference between the first and last term is 40. ...
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### Let $α,β$ be the roots of $x^2-x+p=0$ and $γ,δ$ be the roots of $x^2-4x+q = 0$ where $p$ and $q$ are integers. If $α,β,γ,δ$ are in GP then $p + q$ is?

Let $α,β$ be the roots of $x^2-x+p=0$ and $γ,δ$ be the roots of $x^2-4x+q = 0$ where $p$ and $q$ are integers. If $α,β,γ,δ$ are in GP then $p + q$ is ? My solution approach :- I have assumed $α,β,γ,δ$...
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### I have decided to study math on my own and I came across these questions. Would really appreciate if anyone could help and explain how to solve these. [closed]

I could not come up with a solution for both of the problems, so help is appreciated
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### 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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### 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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### 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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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 $$\frac{\sum_{i=0}^{2019} 3^i}{3^4} = \... 1 vote 2 answers 378 views ### (ab+bc+ca)^3=abc(a+b+c)^3, prove that a,b,c are in G.P. [duplicate] Suppose a,b,c are non-zero real numbers such that$$(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)$... 2 votes 2 answers 89 views ### One root common to$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....
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: ...