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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A question about geometric series/progressions

Is it possible to find the value of the common ratio $r$ given the first term and the sum to $n$ terms without using a numerical approach and solving analytically? In other words, can I simplify $$\...
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Looking for an expansion on the AP sum formula

If I have an x where x starts at x=5, and each step adds 10, so that x1=5, x2=15, x3=25, etc...so that if there were 3 steps the answer would be 5+15+25=45. This is most properly answered by https://...
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Analytical expression for tetrahedral progression

During my engineering studies we did get some Calculus and Algebra background, but I have a lack of knowledge in other topics such as Combinatorics, Recurrences and Progressions. Therefore I would ...
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What is the correct name for the "power function progression"?

I'm trying to find a technically correct name for the following progression $n_{cur} = n_{prev}^2$ It does not look like a geometric progression, or maybe I do not understand it correctly? Also, how ...
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Generalised formula for the given series

I have the below series : (1 * 0) + (2 * 1) + (3 * 2) + (4 * 3) + ... + (n * (n-1)) Is it possible to have a generalised formula for this. Also such series like ...
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Angles of a triangle - are in a Geometric Progression, possible values for the common ratio other than 1 [closed]

Let us assume that there exists a triangle with measures of its angles in a Geometric Progression (G.P.) with a common ratio other than 1. Then what are the possible ranges of (that is starting set ...
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way to distinguish of three lines are radiating from one another

I am backtesting on moving average, and I have to check against the condition of three moving average lines radiating from one another like shown in the picture below. Since I only have the numbers ...
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"Magic" numbers are those divided by all partial digit sums: prove that there is no infinite set of "magic" powers among the natural powers of $\ell$

For a natural number $n$, let $P_n$ the set of sums of each subset of digits in decimal notation of $n$. A number is magic if for each $s \in P_n$, we have $s \ | \ n$. Let's consider a number $\ell$, ...
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Succession of geometric shapes

A succession of geometric shapes is obtained by dividing squares into smaller squares. The first three geometric shapes of these successions are illustrated as follows in the figure: Considering that ...
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Possible values of r (common ratio) if r is equal to d (common difference)

The common difference d of an AP is equal to the common ratio r of a GP. I have been told that the sum of the first ten terms of the AP is equal to fifteen times the sum of the first three terms of ...
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What is the solution to this problem (Geometric and Arithmetic progression)?

Numbers $a , b,c , 64$ are consecutive members of a geometric progression. Numbers $a,b,c$ are respectively the first, fourth, eighth members of an arithmetic progression. Calculate $a + b - c$
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How to prove that a spiral that I have is logarithmic or archimedean?

I am conducting a research on modelling a spiral.. I know that the shape of the spiral on my pencil shavings is logarithmic indeed, How do i prove that? How do I prove it is logarithmic and not ...
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Find all $a,b$ for which the polynomial has real roots and are in geometric progression.

Find all $a, b$ such that the roots of $x^3 + ax^2 + bx − 8 = 0$ are real and in a geometric progression. I did deduce the answer till $a=\dfrac{-b}{2}$. Using the Vieta's relations I deduced that if $...
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4 answers
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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....
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calcuate $\sum_{i=0}^{n} 2^{2i}$

I want to calcuate this problem: $\sum_{i=0}^{n} 2^{2i+5}$ I know that we can expand this problem like this: $\sum_{i=0}^{n} (2^{2i+5})$ $=\sum_{i=0}^{n} (2^5 \times 2^{2i})$ $=\sum_{i=0}^{n} (32 \...
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If $\tan(\pi/12 -x),\tan(\pi/12), \tan(\pi/12 + x)$, are 3 consecutive terms of a GP then sum of the solutions in $[0, 314]$ is $k\pi$. What is $k$?

if $\tan(\frac{\pi}{12} -x),\tan(\frac{\pi}{12}), \tan(\frac{\pi}{12} + x)$, in order are all three consecutive terms of a GP then sum of all the solutions in $[0, 314]$ is $k\pi$. find the value of $...
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-1 votes
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If $a, b, c, d$ are in G.P., prove that they are in the same place as if they were in a different place.

If $a, b, c, d$ are in G.P., prove that $\left(a^{2}+b^{2}\right),\left(b^{2}+c^{2}\right),\left(c^{2}+d^{2}\right)$ are in G.P. And in general, If $a, b, c, d$ are in G.P., prove that $$ \left(a^{n}...
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1 answer
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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: ...
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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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If $(b + c) , (c + a) , (a + b)$ are in H.P. then find the relation between $\dfrac{a}{b+c} , \dfrac{b}{c+a} , \dfrac{c}{a+b}$ .

If $(b + c) , (c + a) , (a + b)$ are in H.P. then $\dfrac{a}{b+c} , \dfrac{b}{c+a} , \dfrac{c}{a+b}$ are in $(i)$ A.P. $(ii)$ G.P. $(iii)$ H.P. $(iv)$ None of These. What I Tried:- I have that $(b+c),...
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Summation of a finite sequence

This question is linked from my previous question: Summation of a sequence? Given the sequence: $$ a_n = 0.9^{n-1}a_1(1+d+d^2+...d^{n-1}) $$ and $a_1=100$ , $d= 1.5$ How to form an equation to find: $$...
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Convergence of geometric mean over arithmetic mean $G_n / A_n$

Fix $a,d\in \mathbb{R}$ an consider the arithmetic sequence $x_n = a,a+d,a+2d,a+3d,...$ ($x_n = a + (n-1)d $ for each $n$). Now consider $$ A_n = \frac{x_1 + x_2 + \cdots + x_n}{n} \quad \text{ and }\...
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Finding the area of a fractal with geometric sequences.

Construct an infinite fractal. Stage 0 is a unit square. At each stage, a square is appended to the vertices of the previous stage such that the sides are 1/2 the sides of the previous stage and ...
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What properties does a radix in the context of ISO/IEC 7064 pure check digit system require?

ISO/IEC 7064:2003 defines so-called pure check digit systems. The term “pure” refers to only requiring one modulus as opposed to two moduli (p. 2). A character string satisfies the check of a pure ...
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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 ...
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Proving sum of complex roots is zero

Let $\alpha$ and $\beta$ be the complex roots of $z^3 =1$. Show that the sum of the first $n$ terms in the series $1 + \alpha + \alpha^2 +...$ is either $0, 1$ or $-\alpha^2$ depending on the ...
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Find sum of $ar^0 + ar^1 + ar^2 + \dotsm + ar^n$

I am trying to deduce the formula of sum of $n$ terms of a GP in a way not described in the book and hence after taking $a$ as common factor, we are left behind with $r^0 + r + r^2 + r^3$ ( I took $n =...
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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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1 answer
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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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8 votes
2 answers
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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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0 answers
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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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2 votes
5 answers
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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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-4 votes
2 answers
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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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3 votes
1 answer
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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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1 answer
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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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2 answers
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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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1 answer
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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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1 vote
2 answers
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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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1 vote
2 answers
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$(ab+bc+ca)^3=abc(a+b+c)^3$, 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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2 votes
2 answers
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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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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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