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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 calculate an ever increasing number of rounds?

Say you are making a roulette bet and you want to double your bet every time you lose in an attempt to recover what you lost. So, if you lose repeatedly you'd have spent: first round = $10$, second round = $10\times 2$, third round = $10\times…
Luis Novo
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derivation of geometric series summation rule?

The sum of an infinite geometric series can be solved with the below equation, given that the common ratio, $r$, is bounded $ -1
jbuddy_13
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In a geometric sequence $u_1=125$ and $u_6 =1/25$

In a geometric sequence $u_1=125$ and $u_6 =1/25$. a) Find the value of $r$ (the common ratio). b) Find the largest value of $n$ for which $S_n < 156.22$. c) Explain why there is no value of $n$ for which $S_n > 160$. Can someone please help me…
Tamarus Darby
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Geometric series word problem: Number of games played

If a tournament consists of 64 teams playing sudden death games until 1 team is declared the winner. How many games, in total, are played during this tournament? My approach: Teams: 64, 63, 62, ....., 1 Two teams play 1 game, therefore, all together…
Simran
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How do you apply finite geometric series in order to determine the distance a bouncing ball travels up and down at the 10th bounce?

My question is very similar to this one, except I would like to determine the distance traveled immediately after the 10th bounce. Assume the ball is let go from 1 meter above the ground and each successive bounce is 2/3 the height of the previous…
user1068636
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Is ${s_n} = \sum\limits_{i = 1}^n {\dfrac{1}{{{3^{i - 1}}}}} = \dfrac{3}{2}\left( {1 - \dfrac{1}{{{3^n}}}} \right)$?

Here in example 4 this result is shown. ${s_n} = \sum\limits_{i = 1}^n {\dfrac{1}{{{3^{i - 1}}}}} = \dfrac{3}{2}\left( {1 - \dfrac{1}{{{3^n}}}} \right)$ Now without looking at its solution I was trying like this: ${s_n} = \sum\limits_{i = 1}^n…
Daman
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How do you find the common ratio of a geometric sequence if not given the first term?

The only given values are the sum of an infinite geometric series which is equal to 9/2, and the second term which is equal to -2. How do I find the common ratio here?
sup
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Sum of a Geometrico-geometric series

Is there a closed form solution for the sum of an infinite geometric series, where the growth from element n to element n+1 converges to zero for $n\rightarrow\infty$? Specifically, I'm looking for a closed formula for…
DanielH
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What kind of geometric series is $f(m) = \sum_{r=1}^{m}\frac{r+1}{n-r-1}$?

I have a geometric series that looks like $$ f(m) = \sum_{r=1}^{m}\frac{r+1}{n-r-1} \\ = \frac{2}{n-2} + \frac{3}{n-3} + \frac{4}{n-4} + \cdots + \frac{m+1}{n-(m-1)} \\ $$ I wrote out a few terms to see if I can identify a simplification but I…
roulette01
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Cash shortfall calculation

Suppose I have a business that has expenses totaling 200,000 per month. Currently its revenue is 100,000 per month. The revenue is growing 10% per month. We can calculate that the business will reach profitability using (log(200000 / 100000) /…
Finbarr
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Applying geometric sum formula in context with indicator function

Consider the following: $$\sum_{k=0}^{n-1} v^k \mathbb{1}_{ \{K \geq k\}} \overset{!}{=} \frac{1-v^n}{1-v} $$ $K=0,1,2,3,.......$ is a random variable. I do not think that I can apply the geometric sum formula here. I do not know how many summands…
Sarah
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Find formula for geometric series with two(?) variables.

I'm trying to find a formula for the series below without success. I've tried to use: $\sum_{k=1}^n ar^{k-1} = \frac{a(1-r^n)}{1-r}$ but my $n$ messes things up. I have trouble understanding what is the first term and the ratio in my series. Any…
Lars Logik
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If the $n$-th term of a geometric progression $5,-\frac 52, \frac 54$.. is $\frac{5}{1024}$, then n is

Common ratio = $r=\frac{-1}{2}$ First term=$a$ $$\frac{5}{1024}=5\left(-\frac{1}{2}\right)^{n-1}$$ $$\frac{1}{1024}=\left(-\frac{1}{2}\right)^{n-1}$$ $$1024=(-2)^{1-n}$$ $$(-2)^{10}=(-2)^{1-n}$$ Then $$10=1-n$$ $$n=-9$$ which makes so sense. How…
Aditya
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Proving that the sum of a geometric series is of the order of its last term

How can we prove that the sum of a geometric series is of the order of the last term for r>1, that is to show that its theta notation consists of the last term.
user_LR
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How to find the Sum of a geometric sequence in the case where the sum does not start at k=0

Let ($u_n$) be the sequence defined by $u_0 = −2$ and $∀n ∈ N, u_{n+1} = 5u_n$ What is $u_{25} + u_{26} +· · · + u_{35}$ ? I have been studying geometric sequences in the last couple of days but this is the first time I've come upon a problem…
Riona Sopi
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