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Questions tagged [geometric-series]

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

A geometric series is of the form

$$\sum_{n=0}^{\infty}ar^n.$$

If $|r| < 1$, then the series converges to $\frac{a}{1-r}$. If $|r| \geq 1$, the series diverges.

886 questions
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How to isolate the common ratio from a finite geometric series formula.

For the formula $T = \frac{a\left(1-r^n\right)}{1-r}$ how do you isolate $r$ in terms of $T$, $a$, and/or $n$?
John Yu
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Explanation of finding the 4th term of a Geometric Sequence

I completed a problem but it seems as if I got the wrong answer. I would like to see what error I made so I do not make the same mistake again. The questions goes as follows : "If the sum of the first $n$ terms of a geometric series is given by…
user55614
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Geometric series: please explain the process for the formula

I get how the formula is made. What I don't get is why whoever decided to multiply the equation by $(1-r)$. I'm referencing this link at Wikipedia. I get the mechanics, just not some of the logic. Consider $$ \sum_{i=0}^{n-1}…
Andrew Falanga
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Geometric Sequence Question - Verbal

"An empty pool is filled with water within 5 hours at a decreasing speed. The amount of water filled each hour is a constant part of the amount that was filled the hour before. The amount of water in the first 4 hours is twice the amount in the last…
Paula
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Calculating infinite sums geometric series

It has been a while since I've dealt with infinite sums and geometric series. Was wondering if anybody could shed some light on this and help me out. $ \sum_{r=1}^\infty \sum_{t=r+1}^\infty 0.02(0.9)^{r-1}(0.8)^{t-1} $ $=\sum_{r=1}^\infty…
Matt S
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Inex shift unclear in proof

Here, at point 5, you can find a proof that, for a variable with a geometric distribution, $P(N>n) = (1-p)^n$ The proof is a follow: $P(N>n) = \sum_{k=n+1}^{+\infty}P(Y=a)=\sum_{k=n+1}^{+\infty}(1-p)^{k-1}p= \frac{p(1-p)^n}{1-(1-p)}=(1-p)^n $ I am…
Jérôme Renaux
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Converting to a familiar form

$(-1)^n × 2^{1/n}$ Is it possible to convert this into the form $ar^{n-1}$? I am not so sure on how to convert this. Can someone give me hints or someone guide me in solving this problem. Additionally, if I were to test this series (summation from…
salierii
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Geometric series with binomial coefficients

Please sum this sum , $\sum_{k=0}^{n} \binom{r+a+bk}{a+bk} x^{bk}$. Where r ,a and b are fixed integers
Brajesh Kumar Dhakad
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Help with finding the number of terms in a geometric sum

I'm currently stuck on a problem involving a geometric sum, and I was hoping to get some assistance with it. Here's the problem: A geometric sum is equal to 215. The first term is 5, and the last term is 320. I need to find the number of terms in…
Bishop_1
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If $S=1+\frac{1}{\sin x}+\frac{1}{\sin^2x}+...$ what's the measure of angle $x$?

If $S=1+\dfrac{1}{\sin x}+\dfrac{1}{\sin^2x}+...+\dfrac{1}{\sin^n x}+...=\dfrac23,$ what's the measure of angle $x$? We have an infinite geometric series with first term $a_1=1$ and common ratio $q=\dfrac{1}{\sin x}$. Then…
yinivem462
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Stuck while solving a question about geometric sequence

I was trying to solve this question from the book Why math by R.D Driver: Imagine a large piece of paper five-thousandths of an inch thick being torn in half, and the two pieces placed one on top of the other. Then these two pieces are torn in half,…
James
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geometric series with negative exponent

I wonder if it is possible to convert the infinite geometric series with negative exponent to a positive one? Is the calculation is correct ? ∑_(n=1)^∞▒x^(-n) =∑_(n=1)^∞▒〖(〖1/x)〗^n 〗=1/(1-1/x)=x/(x-1) The pic is attached.
Neda Fathi
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2 cars and the fly problem - calculate the total distance covered by the fly

2 cars approach each other, with 20 km between them. The speed of each car is 10 kmph. At 20 km apart from each, a fly starts traveling from one car towards another at 15 kmph. Once it reaches the other car, it turns back and starts towards the…
Shiro Kumo
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Is the Geometric Series defined at x=0?

The geometric series is usually defined as $\sum_{k=0}^{\infty} a \cdot x^{k}$ where $x$ is on the interval $]-1;1[$, which includes $0$. My Problem is that substituting $x=0$ for the first Term of the sum gives $a \cdot 0^{0} $. This is an…
Qwox
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"modified" geometric series

Good evening, Let $n, b \in \mathbb{N}$, $n,b \geq 2$ and $b \in \mathbb{N}$, then we know that $\sum_{k=0}^n b^k = \frac{b^{n+1}-1}{b-1}$. In other words we have $\sum_{k=0}^n b^k = c_{1,b} \cdot b^n + c_{2,b}$, where $c_{1,b},c_{2,b}$ are…
student7481
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