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Questions tagged [geometric-series]

For questions about or involving geometric series, a series where successive terms have a common ratio.

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Conditional probability greater than 1. Why?

Let's say I roll a fair die independently many times. Let $X_i$ be the outcome of the $i$th roll. Assume that on the $k$th roll, I get a $1$ or $X_k = 1$. What is the probability that I never rolled ...
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2answers
60 views

Why do these expression tend to two? [on hold]

Sorry if I express myself wrongly... Why in a formula like this $x^n-x^{n-1}-x^{n-2}-x^{n-3}-x^{n-4}-...-x^{0}=0$ with $n \to \infty$ $x \to 2$ ?
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2answers
42 views

Sum of (Almost) Infinite Geometric Series

I have recently stumbled upon this old problem of proving that $$ \displaystyle\sum_{k=0}^{\infty} \left\lfloor\dfrac{n+2^k}{2^{k+1}}\right\rfloor=n, \, \forall n\in \mathbb{Z}_+ $$ where $\left\...
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0answers
41 views

Non constant geometric sequence problem?

A problem on the national maths competition which I can't solve no matter how hard I try. It's high school maths so it shouldn't be that difficult. So here is the question: How many prime numbers can ...
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1answer
39 views

Formula for this diminishing function

I'm trying to write a function for a game where points accumulated becomes less effective as they get more. Input Value: 2000 Interval: 500 For every interval reached, decrease the ...
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1answer
35 views

Compute the Sum of geometric series

Basically, I just came out of my exam and this was a question that I wasn't sure of how to solve, I'd appreciate it if someone answered this $$\sum_{k=1}^{99}\frac{1}{k(k+1)}$$
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2answers
27 views

Solve geometric series equation with large terms

Let $\{a_i\}$ is a geometric sequence with common ratio $r=2/3$. If $a_1+a_2+...+a_{100}=15$, $a_1+a_2+...+a_{99}$? I think $a_1(1+\frac{2}{3}+...+(\frac{2}{3})^{99})=15 \implies a_1=5$, what wrong?
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1answer
25 views

Incorrect derivation of geometric series

So this is something really really simple but for some reason I honestly cannot figure out why this is wrong. I was deriving the equation of the summation of a geometric series to the nth term ...
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5answers
795 views

Two players alternate flipping a coin until the result is head. How to derive that the probability for the first player to win is $2/3$? [duplicate]

Two players, $A$ and $B$, alternately and independently flip a coin and the first player to obtain a head wins. Player $A$ flips first. What is the probability that $A$ wins? Official answer: ...
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3answers
32 views

Looking for the Closed Form of a Two-Variable Geometric Sum

Is there any closed form of the equation $$\sum_{i=0}^n a^{n-i} \cdot b^i$$ for real values $a$ and $b$ and integer $n \ge 0$?
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2answers
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Proof for a prime number equation

Let x and n be positive integers such that $$1 + x + x^2 + x^3 ... + x^{n-1}$$ is a prime number. Then show that n is a prime number. So I have summed up the GP and equated it to y which is a prime ...
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2answers
57 views

Rational as series?

I was checking out a few things in the geometric series and realized all rational numbers can be shown as a geometric series.I was pretty sure I read something like that somewhere.Can anyone tell me ...
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3answers
575 views

What is the maximum value of $\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$?

If $x$ is positive, what is the maximum value of this expression: $$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$ This question is from a book of problems on sequence and series under the section on AM-...
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0answers
31 views

Transforming sum of phases to geometric series of sine function

While trying to understand Shor's algorithm, I encountered this formula: $p(y) = \frac{1}{2^nm}\big|\sum^{m-1}_{k=0}e^{2\pi ikry/2^n}\big|^2$. Now for $y=j2^n/r+\epsilon$, according to my textbook, ...
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1answer
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How do I prove this relationship between positive terms of a G.P.?

$a$, $b$, $c$, and $d$ are positive terms of a G.P. This is the relationship I'm trying to prove: $$\frac1{ab} + \frac1{cd} > 2 \left(\frac1{bd} + \frac1{ac} - \frac1{ad}\right)$$ This ...
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2answers
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I have an inequality which is giving me the correct answer, except that the sign is opposite to what it should be, any help?

I'm looking at a geometric series that looks like $A_n = \sum_{k=0}^{n}A_0x^k$ where $A_0$ is a constant. I'm looking to find an $n$ where my error between two terms is less than a certain value $E$...
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6answers
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Is $1111111111111111111111111111111111111111111111111111111$ ($55$ $1$'s) a composite number?

This is an exercise from a sequence and series book that I am solving. I tried manipulating the number to make it easier to work with: $$111...1 = \frac{1}9(999...) = \frac{1}9(10^{55} - 1)$$ as ...
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0answers
25 views

Optimizing a weight algorithm

I am attempting to optimize a C++ algorithm that calculates a set of values with corresponding weights. There are n values, with the first being the most recent and ...
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2answers
39 views

Which positive real number x has the property that x, floor of x and x - floor of x form a geometric progression

Which positive real number $x$ has the property that $x$, $\lfloor x \rfloor$, and $x - \lfloor x\rfloor$ form a geometric progression (in that order)? (Recall that $\lfloor x\rfloor$ means the ...
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4answers
454 views

Expression for sum of $n$ exponentials

So I have this sum of exponentials and I would like to find an expression for it. $$\sum^n_{i=1} e^{\mu(i-1)} $$ Note that $i$ is not an imaginary indicator. I am aware there is a formula for ...
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2answers
66 views

Prove that every geometric sequence allows for $S_n(S_{3n}-S_{2n})=(S_{2n}-S_{n})^2$

$S_n(S_{3n}-S_{2n})=(S_{2n}-S_{n})^2$ Is there a way to prove this without expanding everything based on the geometric sum formula? I get lost very easily when trying to solve this conventionally and ...
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2answers
50 views

Convert expression to standard geometric progression

How can I transform a formula describing a curve (a discrete series) to the standard formula for a geometric progression? (Note: Maths is still very much a foreign language to me. I could be using ...
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2answers
30 views

Finding the infinite sum of a geometric series

I have the series: $3 + \frac{3}{2} + \frac{3}{4} + \frac{3}{8}$, I need to calculate the exact value of it. I have found that the equation of this is $a_n = 3(\frac{1}{2})^{n-1}$, but I am not ...
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1answer
24 views

Find the Year of the Question paper it belongs to (Maths 9709 ) [closed]

The first two terms of a geometric progression are where 0<θ<π/2 (i) Find the set of values of θ for which the progression is convergent. [2] Which Year Question paper is this? Year with ...
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0answers
114 views

Show $ \sum_f \lambda(f)t^{\deg f} = \prod_g \big(1 - \lambda(g)t^{\deg g}\big)^{-1} $

So I want to show that $$ \sum_f \lambda(f)t^{\deg f} = \prod_g \big(1 - \lambda(g)t^{\deg g}\big)^{-1} $$ where the sum is over monic polynomials and the product is over all monic irreducible ...
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1answer
28 views

Me possibly overthinking a geometric series question

I am creating a study sheet for geometric series and came across this questions Do you think there is an infinite geometric series with first term 10 and a sum of 4? If so, find one infinite ...
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2answers
42 views

Sum of infinite geometric series with two terms in summation

I have an infinite geometric series: $$\sum_{i=1}^{\infty} \frac{r^i}{(1+d)^i},$$ where $d$ is a constant. I would like to use the sum of infinite geometric series formula, but I cannot see how to use ...
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0answers
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Computing the value of a function based on the geometric series of 1/2^n

So in John B. Conway's First Course in Analysis Book we're told to consider the interval $[a,b],$ and let $\{r_n\}$ denote a sequence of all rationals in said interval. Next, we define a map $\alpha:[...
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1answer
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Complex geometric series $\frac{1}{6i}\sum_{k=0}^{\infty}\left(\frac{z-3i}{6i}\right)^n$

$\dfrac{1}{z+3i}$ can be interpreted as the sum of the geometric series $\displaystyle\dfrac{1}{6i}\sum_{k=0}^{\infty}\left(\frac{z-3i}{6i}\right)^n$ this can be obtained by writing: $\dfrac{1}{z+3i}=\...
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0answers
25 views

Unit step function into geometric series

The unit staircase function is defined as follows: $f(t)=n$ if $n-1 \le t < n$ for $n \in \{1,2,3, \ldots\}$. How do you show that $\displaystyle f(t) = \sum_{k=0}^\infty u(t-k)$?
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2answers
37 views

$\frac{1}{x^2} $ using geometric series vs. Taylor

So I have the function $f(x) = \frac{1}{x^2} $ and I want to represent it as a Taylor series [centered @ x=1] (also evaluating at $x = 1.02$). $$$$I did it using the standard Taylor series method: $...
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0answers
33 views

Why are these 2 summations not equal to each other?

Why is:$$\sum_{n=1}^\infty \left(\frac{(-1)^{n-1}(x-1)^n}{n}\right) \neq \sum_{n=0}^\infty \left(\frac{(-1)^{n}(x-1)^{n+1}}{n+1}\right)$$ This question arose when I tried to get the series ...
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0answers
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Finding values for range of terms in a geometric sequence

I am NOT math person. I'm a musician who needs some math help. Given the following: A1 = An / (r)^(n-1) I know "An" and "r" and "n"; I want to find "A1" but only for a specific range of terms in ...
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4answers
112 views

Solve recurring sequence using a generating function

I have the sequence $a_n=3a_{n-1}-3a_{n-2}+a_{n-3}$, $\forall\ n \ge 3$, with $a_0=2$, $a_1=2$, $a_2=4$ being the known terms, and I want to find a non-recursive equation for $a_n$ using a generating ...
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1answer
52 views

How can I simplify this expressions to get one formula?

the expressions are : f(1) = (a^0) f(2) = (a+1) f(3) = (a^2+a+1) and the answer is f(n)= (a^n-1) /a-1, it is the formula for the sum of the geometric series right ? I have tried to find the formula ...
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2answers
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Prove that $\sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty$, where $|r_1|, |r_2| < 1$

I'm trying to prove that: $$ \sum_{j=0}^{\infty}|c_1r_1^j + c_2r_2^j| < \infty $$ where $|r_1|, |r_2| < 1$ and $c_1, c_2$ are some arbitraty real numbers ($r_1, r_2$ are also real numbers). If ...
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1answer
37 views

Geometric series, TMUA exam question, please help!

Q. A geometric series has first term $4$ and common ratio $r$. Where $0 < r < 1$ The first, second and fourth terms of this geometric series form three successive terms of an arithmetic ...
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1answer
41 views

How can I solve this geometrical sequence problem?

How can I solve this problem and can you explain it? $S_n$ is the sum of the first $n$ terms of a geometric sequence with first term $a$ and a common ratio $r$. Let $P_n$ represent the product of ...
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Sum - geometric series [closed]

I have series a + 2(1-a)a + 3 (1-a) (1-a)a + 4 (1-a)(1-a)a + … Does this series have a common ratio? How can I calculate sum and average?
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1answer
72 views

How to calculate the summation of $n \cdot 2^n$? [duplicate]

So I know that you can take the derivative of this and multiply by x and do integrals or something like that. However, I am just wondering if there is a way to come to the summation of the series ...
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1answer
30 views

Substitution in Infinite Geometric Series

I'm getting two different answers for the following series... $$\sum_{k=0}^{\infty} q^{2k}$$ With $-1 < q < 1$, I can solve the series by using the infinite geometric sum formula. $$\sum_{k=...
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0answers
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Nth term of geometric series where each term is rounded to two decimal places

I know that the nth term of a geometric series is $$f(n, a_1) = a_n = a_1 r^{n-1}$$ In my case however, since I am dealing with money, I am rounding each term to two decimal places. I calculate the ...
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0answers
41 views

I've a math problem about series but can't find a suitable formula

I'm developing a program that requires a mathematical formula. You can see it below: "1+1.1+1.21+1.331+1.4641+1.6105+1.7715……" The problem is that I want to find the sum of 100 or 1000 numbers in ...
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3answers
33 views

Summation of a geometric sequence from $1$ to infinity for: $(n^{2})\times ((\frac{5}{6})^{n-1})$

I'm fully comfortable with most series and even arithmetico–geometric sequence including n to any exponent if the geometric term is in the form of $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$, and so forth....
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3answers
52 views

Determine the value of $\sum_{k=0}^n i^k$

this is the problem: Determine the value of the sum $\sum_{k=0}^n i^k$ for $n \in \Bbb N$. I started to sum term by term but don't get its value
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0answers
83 views

Can it be said that $\sum_{k=n}^{\infty} q^{k} = \sum_{k=0}^{\infty} q^{n} \cdot q^{k}$?

I would like to know if the following equality is always true for geometric series. $$\sum_{k=n}^{\infty} q^{k} = \sum_{k=0}^{\infty} q^{n} \cdot q^{k}$$ It has worked for me on several occasions, ...
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3answers
61 views

Finding a geometric series for $\frac{1}{(1 + x)^2}$

I'm asked to find a geometric series for $f(x) = \frac{1}{(1 + x)^2}$, I integrate it first and I get $F(x) = \int{\frac{1}{(1 + x)^2}dx} = \frac{1}{-2(1 - (-x))}$ Since $\frac1{1-x} = \sum_{n=0}^\...
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1answer
29 views

Prove that$\>$for $x\neq1$ and k ${\in}\> \textbf{Z}_{\geq0}$,$\>\sum_{j=0}^{k}x^{j}=\frac{1-x^{k+1}}{1-x}$.

My response was as follows: We may test our base case, that is when $k=0$. This computes to $x^{0}=\frac{1-x^{1}}{1-x}$ which equates to $1=1$. Whence $k=0$ holds. Therefore, such a statement $P_{k} ...
0
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1answer
36 views

Geometric series of entries of matrix with spectral radius smaller

I have a matrix $A$ such that the spectral radius is $< 1$. It is well known that $I+A+A^2$... converges. Does it then follow that the geometric series of each entry also converges? The matrix ...
3
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1answer
99 views

Series of Squares Under Triangle

A line crosses the $x$ and $y$ axes at $(a,0)$ and $(0,1)$ respectively, where $a>0$. Square are placed successively inside the right angled triangle thus formed. What is the area enclosed by all ...