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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Alternate proof of divisibility of $(a^n - b^n)$ by $(a - b)$
I know that $a-b$ can divide $a^n-b^n$ and I have already seen other proofs in a similar post here. But, I want to know if the following way of proving it is correct or not.
Say we have a G.P. : $1, \...
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Prove that $T_{n}$ is the $n$th term of a GP.
Let $a, b, c$ be some constants, and $T_{n}, S_{n}$ be the $n$th terms of some (different) sequences such that:
$$
\begin{align}
T_{n} & = \cfrac{2a(b-2S_{n})}{(a+c)^{2}}
\\ S_{1} & = 0
\\ S_{...
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Can sequence 1, 1, 1, 1,.. be called harmonic sequence (beside being geometric and arithmetic)? [closed]
Sequence 1,1,1,1 or for example 5,5,5,5 is arithmetic and geometric, but can it also be harmonical or some other type of sequence?
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Find where the product of terms of a geometric progression reaches its maximum [closed]
The geometric progression $\{a_n\}$ has first term $a_1 = 1536$ and common ratio $-\tfrac12$. What is the value of $n$ for which the products of the first $n$ terms is maximum?
I really do not ...
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G.P. geometry relation
By means of a single geometrical diagram, can you show why a geometrical progression is called a geometrical progression?
I have thought about showing consecutive diagrams of square areas to either ...
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$\frac{a}{x},\frac{b}{y},\frac{c}{z}$ are in HP
If non-zero numbers $a,b,c,x,y$ and $z$ are such that $a,b,c$ are in AP, $x,y,z$ are in GP and $\frac{a}{x},\frac{b}{y},\frac{c}{z}$ are in HP then prove $|a|=|c|$.
I have been trying to solve this ...
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Write the geometric progression equation
There is a sequence whose elements have the following dependency
a[i] = (a[i-1] + 2 * a[i-1] * Sqrt(2)) / 2
a[i=1] = 10
I need to write a sequence equation and ...
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Arithmetic-Geometric progression general integral formula
I'm solving Arithmetic-Geometric progression. It's rules:
$$
k(x)\in Z, \forall{x}\\
f(0)=k(0)\\
q(x) > 0, \forall{x}\\
f(x)=f(x-1)\frac{q(x)}{q(x-1)}+k(x)-k(x-1)
$$
I got the general formula:
$$
f(...
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Perfect Square from Geometric Progression
This question is from QuantGuide(namely Geometrical Progression):
Write out a series of whole numbers in geometrical progression with at least 3 terms, starting from
1, so that the numbers add up to a ...
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Is there a way to find the common ratio with the sum and first value of a geometric sequence?
Given a geometric sequence $U(n)$, its only known values are $U_{0}$, $S$ the sum of the geometric sequence till a certain $N$-th term.
Is it possible to find $q$ via the following sum equation:
$$
S =...
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Running a Ponzi (game theory).
Let's say I'm organizing a transparent Ponzi scheme for gambling purposes.
It's a game of doubles. Each player's goal is to double their deposit. Each new entree pays for the players before him in ...
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Proof of conditions for polynomials
Find the conditions for the roots $\alpha, \beta, \gamma$ of the equation $x^3-ax^2+bx-c=0$ to be in: $(i)$A.P.; $(ii)$G.P.
If the roots are not in A.P. and if $\alpha+\lambda,\ \beta+\lambda,\ \...
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Calculate the dimensions that are in progressive progression.
Give the surface and diagonal of a parallelepiped rectangle. Calculate the known dimensions that are in progressive progression.
(Answer: $\dfrac{2d^2+S-\sqrt{(2d^2+3S)(2d^2-S)}}{4\sqrt{d^2+S}}; \...
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Sum of geometric progression [closed]
Given a geometric sequence whose sum of the first ten terms is 4, and whose sum
from the 11th to the 30th term is 48. Find the sum the 31st to the 60th term.
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Find the sum of the expression below [duplicate]
Calculate:
$\dfrac{1}{2}+\dfrac{2}{2^²}+\dfrac{3}{2^3}+\dfrac{4}{2^4}+...=?$
I try
$\dfrac{1}{2}+\dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{4}+\dfrac{1}{8}+\dfrac{1}{16}+...\dfrac{1}{8}+\dfrac{1}{16}+\...
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Divergence of the sum of the reciprocals of a subset of the prime numbers
I had the following question related to the sum of the reciprocals of the prime numbers restricted to a certain subset.
Let $T, c > 1$ be real numbers. Let $A=\cup_{k \geq 0} [Tc^{2k},Tc^{2k+1}]$ ...
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The cubic $x^3+ax^2+bx+c$ has three distinct zeros in GP and the reciprocal of these zeros are in AP then prove that $2b^2+3ac=0$
I tried to solve this question: first assuming the zeros to be
$m$, $mr$ and $mr^2$ in G.P.
so that
$\frac{1}{m}$, $\frac{1}{mr}$ and $\frac{1}{mr^2}$ in A.P.;
then by solving it like
$\frac{1}{mr}-\...
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Prove that $0.\overline{123} = 0.123123123123...$ is rational [closed]
How are we to go about proving that $0.\overline{123} = 0.123123123123...$ can be expressed in the form $\frac{p}{q}$ where $p,q \in \mathbb{Z}$, i.e is rational?
Is this to be done using arithmetic ...
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If $f(x)=2+6x+18x^2+54x^3+\ldots$ and $f(x) = O(g(x))$ [closed]
If $f(x)=2+6x+18x^2+54x^3+\ldots$ and $f(x) = O(g(x))$, what is the value of $g(x)$ in the above sequence ?
I tried calculating $f(x)$ using the sum of an infinite GP but can't understand how to find $...
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Why do two answers appear here in Sequence and series
The question goes like this , The sum of first three terms of a GP is 13/12 and their product is -1
Find the common ratio and terms of the GP
My answer went something like this :
Assume 3 numbers to ...
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summation of $3^k$ [closed]
how do you write the closed form of a sum of the geometric progression of 3^n? Our teacher told us that $2^0+2^1.... 2^n$ is equal to $2^{n+1}-1$ but I am not sure how to apply that to a similar ...
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Finding the summation of this peculiar type of infinite series
Find $\Sigma_{r=1}^{\infty} \frac{1}{a+\frac{b^{r}}{r}}$ where $b>1$ and $a$ is a positive real number.
I guess that this sum must be convergent as the terms gets smaller, but I have no idea on ...
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$\int_0^1\frac{1}{7^{[1/x]}}dx$
$$\int_0^1\frac{1}{7^{[1/x]}}dx$$
Where $[x]$ is the floor function
now as the exponent is always natural, i converted it to an infinite sum
$$\sum\limits_{k=1}^{\infty} \frac{1}{7^{[1/k]}}$$
Which is ...
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Is it true that if the positive numbers are in A. P or G. P, then the number must be roots of some polynomial equation?
The above question is in reference to the process of solving the following problem.
Question: If the arithmetic means of two positive numbers $a$ and $b$, where $(a>b)$ is twice their G.M then ...
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How to find the union of any progression?
Here is an answer to find the union of two arithmetic progression.
How to find a general formula for union of two arithmetic progressions
But, is there a formula to find the union of the sets of two ...
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Minimum distance between $2^n$ and $3^m$
I’m looking for the minimum distance between any two members of the geometric progressions 2, 4, 8,… and 3,9,27,…
It seems like the pair of numbers which has the minimum distance is (2,3). Can you ...
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Prove the sequence of three real numbers
If $a,b,c$ are non zero real numbers satisfying $$(ab+bc+ca)^3=abc(a+b+c)^3$$ then prove that $a,b,c$ are terms in $G.P$
My work:
I assumed that they are in $G.P$ and so assumed $b=ak$ and $c=ak^2$ ...
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A positive number, it's integral part and decimal part form a geometric progression. Then the greatest integer less than or equal to the number is?
Let the number be $x$.
Integral part of the number $=[x]$
Decimal part of the number $=x-[x]$
Now as per the question $x,[x] $ and $x-[x]$ form a geometric progression. So :-
$[x]^2=x(x-[x])$
Now I ...
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A question about geometric series/progressions
Is it possible to find the value of the common ratio $r$ given the first term and the sum to $n$ terms without using a numerical approach and solving analytically?
In other words, can I simplify
$$\...
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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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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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"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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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....
user876009
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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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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
$$
\left(a^{n}...
user845875
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geometric progression point distribution when one extreme point is negetive
How to generate 20 points from -0.01 to 100 which are geometrically equal in separation means if I want to plot in log scale $\log d_2 - \log d_1 = \log d_3 - \log d_2$ where $d_1$, $d_2$, $d_3$ ...
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Generalized Arithmetico-Geometic series with a possible application
While playing with Arithmetico-Geometric progression formula(i.e
$$\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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0
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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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