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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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Can the formula for a series be "reverse engineered" given some of its terms?

Say I have a sequence that goes: 12, 16, 21, 27, 34, 42, 51 and so on in a manner that the increment between $a_1$ and $a_2$ is 4, between $a_2$ and $a_3$ is is 5 etc. The formula which describes this series is $(n(n+7)/2)+12$. My question is: would…
Fabio Freitas
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What's wrong?: Find the infinite sum $S = 1 + \frac{3}{4} + \frac{7}{16} + \frac{15}{64} + \frac{31}{256} + \ldots$

The answer I got by hand is not the same to the one I found using a spreadsheet. $\displaystyle S = 1 + \frac{3}{4} + \frac{7}{16} + \frac{15}{64} + \frac{31}{256} + \ldots$ $\displaystyle \frac{1}{4}S = \hspace{8.5pt} \frac{1}{4} + \frac{3}{16} +…
asd
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Is it possible to find the nth term and finite sum for a recursive sequence with case involved

Say: \begin{gather} a_1 = 55 \\ \\ a_{n + 1} = \begin{cases} r_a a_n + C, & \text{if $n$ mod 12 is 6,7,8} \\[2ex] r_b a_n+C, & \text{otherwise} \end{cases} \end{gather} where: $C$ is a constant. $0
Chi Chan
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Find the sum of the following geometric series $\sum_{k=2}^{\infty} \frac{5}{2^k} = \frac{5}{2}$

Find the sum of the following geometric series $$\sum_{k=2}^{\infty} \frac{5}{2^k} = \frac{5}{2}$$ Attempt: First I test with the root criterion if its divergent or convergent... $\frac {1}{2} < 1$ so it's convergent... Now I try to find the sum…
Vek
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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. $$\sum_{k=0}^{\infty} q^{2k} = \frac{1}{1-q^2}$$ However, if I make…
Yulmart
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Confusion with the formula to sum the terms of a finite geometric series

The formula to sum the terms of a finite geometric series is the following: $$\frac{a_1(1 - r^{n+1})}{1 - r}$$ where $a_1$ is the first term, $r$ is the common ratio, and $n + 1$ is the number we want to sum up to. Now, my problem is really in this…
user168764
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If there is a probability of (1-gamma) of you dying everyday, what is your expected lifetime?

Study Question: Verify this fact: if, on every day you wake up, there is a probability of $1 − γ$ that today will be your last day, then your expected lifetime is $1/(1 − γ)$ days. I thought lifetime of $n$ days means surviving $(n-1)$ days and…
X Y
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Weird relation that I found: Any relation to Geometric Series?

I was doing some chemistry homework, and I happened to find the following relation between some numbers: $$(-273.15 + 121.4)\times\frac95 = -273.15$$ Weird coincidence, right? You add a number to an unrelated number, multiply that new number by…
Mailbox
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Need help understand some simple algebra

I was looking at some text that involves the following equality, where $z \in \mathbb{C}$ $(1+|z|^2+|z|^4+\dots +|z|^{2n-2})=\frac{|z|^{2n}-1}{|z|^2-1}$ I assumed the text used the formula for finite geometric series but shouldn't that give…
Remu X
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Geometric series from -i to i

Is it possible to make 2 geometric series from $n = -i, -(i-1), -(i-2),...,(i-1),i$ For a finite series I know that a geometric series looks like that, for $x^n$ $ \sum_0^i = \frac{1-x^{i+1}}{1-x}$
happypaticle
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What is the sum of geometric series if r is approaching 1 from left side?

We know there is a rule that geometric series converges when $|r|<1$ , and a formula to calculate the sum as $S_n=\frac{a}{1-r}$ or as $S_n=\frac{a(1-r^n)}{1-r}$. My question is: when I take $r$ infinitely close to $1$ (as $0.999999...$), wouldn't…
Memduh Yılmaz
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Find the number of ways in which we can select three distinct numbers from the set {10,11,12...,100} such that they form a geometric series.

Let common ratio be $r$ and the first term be $a$. Then $ar^2\leq100 \implies r^2\leq10$ It's quite easy if we consider $r$ to be a natural number. For $r=2$, $10\leq a\leq25$ and for $r=3$, $10\leq a\leq11$. We get $18$ values. The problem arises…
Tatai
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Doubling Sequence

What would be the formula for finding the number of sequences needed to reach some value when we know that the start value doubles on each sequence. So, for example for a problem where we know that the start value is $2.5$ and we want to find out in…
Leff
  • 163
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Geometric series function

The geometric series is defined by : $$\sum^x_{k=1} ar^k$$ But is there a function that is an extension of the geometric series, that satisfies : $$f(1)=r$$ $$f(x+1)=f(x)+r^{x+1}$$ That works for all $x \in \mathbb{R},\:x > 1$?
PearlSek
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Geometric Series-Bounce Height of a Ball

A ball is dropped from a height of $20 \, \mathrm{m}$. It rebounds to a height of $16 \, \mathrm{m}$ and continues to rebound to eight-tenths of its previous height for subsequent bounces. Calculate the total distance the ball travels before it…
Samuel BAUMGARTEN
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