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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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I’m confused why this infinite geometric sum is true m

I’m really confused why this is true $$p \cdot \sum\limits_{i=m+1}^\infty (1-p)^{i-1} = (1-p)^m$$ I know the formula for the infinite geometric sum is $$\sum\limits_{i=0}^\infty a r^i = \frac{a}{1-r}$$
user32091
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Geometric series : Find common ration 'r'

Find common ration of a finite geometric series If the first term is 11 and sum of first 12 terms is 2922920. After applying the formula I got $265720 = (1 - r^{12})/(1-r)$ but I don't know how can I solve for it further.
Vikas Verma
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Help me derive a formula for finding a formula for sum of x^n

I wanna find the sum of divisors of a number defined as:$$2^n$$ Let the series be S, $$ so, S = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 +.....2^n $$ Noww, if we factor out the 2 from 2nd term to last term, S can be written like :$$ S= 2^0 + 2(2^0 + 2^1 +2^2 +…
Chandler Bing
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Sum with geometric series

I try to calculate $$\sum_{k=0}^n(-1)^kx^{(2k)}.$$ I know that the sum of $q^i$ from $i=0$ to n equals $(q^{(n+1)} -1)/(q-1)$. I think this should help with my problem. From mathematica i know that my sum should equal $(1+(-1)^n x^{(2+2n)})$…
Arji
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Sum of powers for a non integer

I know that there are formulas for the first n squares,cubes,..., but is there a formula for the first n powers of a single non-integer? For example, $$\sum_{i=0}^{n}1.1^i$$ Does a formulas to compute this sum exist or do i have to manually go in…
Nick Lennon
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Unexpected result while calculation geometric series sum

I have a geometric series like this : $$\sum_{n=-\infty}^{-1}a^{-n}e^{-jwn}$$ When I make $m = -n$ substituion, it becomes this : $$\sum_{m=1}^{\infty}(ae^{jw})^m$$ And when I calculate summation, the result becomes : $ae^{jw}/(1-ae{^jw})$ I…
jason
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Simplify using Geometric series

Can this be simplified using Geometric series? $$\Large\sum_{a=2}^\infty x^a\left[2(pq)^{\frac{a-2}2}+p^2+q^2\right]$$ thanks!
Jonathan
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What is $\sum r^{an^2+bn+c}$ when $|r|<1$?

I was going through the properties of geometric series and the following question came to my mind. What I am willing to know is the following: For $|r|<1$ what is the value of $\sum\limits_{n=1}^\infty r^{an^2+bn+c}$ with $a,b,c\in \mathbb R$ ? I…
KON3
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Geometric Sequences Problem

The formula for geometric sequences is $a = (a1 * r^{n-1})$, right? Why isn't it working for this problem? $400, 320, 256\cdots$ What is the $6$th term? $a_6 = 400\times{4/5}^5$?
John
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Geometric progression problem with solving a system

I have this system: b1 + b2 + b3 = 195 b3 - b1 = 120 b1 + b1*q + b1*q^2 = 195 b1*q^2 - b1 = 120 I have to find $b_1$ and $q$(this is the private member or the progression) The answer of this exercise is $b_1 = 15$ and $q = 3$ also $b_1 = 125$ and…
user202029
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Is there a geometric sequence that contains each of the numbers 1, 2 and 3?

I recently got the question: Is there a geometric sequence that contains each of the numbers 1, 2 and 3? I have tried my best to make a start with the $x_n=ar^{(n-1)}$ theorem but I'm still puzzled! On stack exchange I have also seen this very…
Leo
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Unable to get correctly the last term of finite geometric series.

Insert $13$ real numbers between the roots of the equation: $x^2 +x−12 = 0$ in a few ways that these $13$ numbers together with the roots of the equation will form the first $15$ elements of a sequence. Write down in an explicit form the general…
jiten
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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?
somkiat_t
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How to find partial sum of the geometric series starting from the 2nd term?

The question asks to find the sum of the geometric series, given this: $$\sum_{m=2}^{10} 5^{m-3}$$ I found that the common ratio is $r=5$. The formula that I use to find the sum is $S_n = a\frac{r^n - 1}{r - 1}$. What do I substitute into n and a?
Liana Pavlicheva
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need help with geometric series

could any one give me hint how to solve $$\sum_{n=1}^\infty \frac{3}{(-2)^n} $$ using the geometric series?
DSL
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